Quadratic Equations - PSAT Math
Card 1 of 399
What is the sum of the values of x that satisfy the following equation:
16x – 10(4)x + 16 = 0.
What is the sum of the values of x that satisfy the following equation:
16x – 10(4)x + 16 = 0.
Tap to reveal answer
The equation we are asked to solve is 16x – 10(4)x + 16 = 0.
Equations of this type can often be "transformed" into other equations, such as linear or quadratic equations, if we rewrite some of the terms.
First, we can notice that 16 = 42. Thus, we can write 16x as (42)x or as (4x)2.
Now, the equation is (4x)2 – 10(4)x + 16 = 0
Let's introduce the variable u, and set it equal 4x. The advantage of this is that it allows us to "transform" the original equation into a quadratic equation.
u2 – 10u + 16 = 0
This is an equation with which we are much more familiar. In order to solve it, we need to factor it and set each factor equal to zero. In order to factor it, we must think of two numbers that multiply to give us 16 and add to give us –10. These two numbers are –8 and –2. Thus, we can factor u2 – 10u + 16 = 0 as follows:
(u – 8)(u – 2) = 0
Next, we set each factor equal to 0.
u – 8 = 0
Add 8.
u = 8
u – 2 = 0
Add 2.
u = 2.
Thus, u must equal 2 or 8. However, we want to find x, not u. Since we defined u as equal to 4x, the equations become:
4x = 2 or 4x = 8
Let's solve 4x = 2 first. We can rewrite 4x as (22)x = 22x, so that the bases are the same.
22x = 2 = 21
2x = 1
x = 1/2
Finally, we will solve 4x = 8. Once again, let's write 4x as 22x. We can also write 8 as 23.
22x = 23
2x = 3
x = 3/2
The original question asks us to find the sum of the values of x that solve the equation. Because x can be 1/2 or 3/2, the sum of 1/2 and 3/2 is 2.
The answer is 2.
The equation we are asked to solve is 16x – 10(4)x + 16 = 0.
Equations of this type can often be "transformed" into other equations, such as linear or quadratic equations, if we rewrite some of the terms.
First, we can notice that 16 = 42. Thus, we can write 16x as (42)x or as (4x)2.
Now, the equation is (4x)2 – 10(4)x + 16 = 0
Let's introduce the variable u, and set it equal 4x. The advantage of this is that it allows us to "transform" the original equation into a quadratic equation.
u2 – 10u + 16 = 0
This is an equation with which we are much more familiar. In order to solve it, we need to factor it and set each factor equal to zero. In order to factor it, we must think of two numbers that multiply to give us 16 and add to give us –10. These two numbers are –8 and –2. Thus, we can factor u2 – 10u + 16 = 0 as follows:
(u – 8)(u – 2) = 0
Next, we set each factor equal to 0.
u – 8 = 0
Add 8.
u = 8
u – 2 = 0
Add 2.
u = 2.
Thus, u must equal 2 or 8. However, we want to find x, not u. Since we defined u as equal to 4x, the equations become:
4x = 2 or 4x = 8
Let's solve 4x = 2 first. We can rewrite 4x as (22)x = 22x, so that the bases are the same.
22x = 2 = 21
2x = 1
x = 1/2
Finally, we will solve 4x = 8. Once again, let's write 4x as 22x. We can also write 8 as 23.
22x = 23
2x = 3
x = 3/2
The original question asks us to find the sum of the values of x that solve the equation. Because x can be 1/2 or 3/2, the sum of 1/2 and 3/2 is 2.
The answer is 2.
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If x > 0, what values of x satisfy the inequality _x_2 > x?
If x > 0, what values of x satisfy the inequality _x_2 > x?
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There are two values where _x_2 = x, namely x = 0 and x = 1. All values between 0 and 1 get smaller after squaring. All values greater than 1 get larger after squaring.
There are two values where _x_2 = x, namely x = 0 and x = 1. All values between 0 and 1 get smaller after squaring. All values greater than 1 get larger after squaring.
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The product of two consecutive positive multiples of four is 192. What is the sum of the two numbers?
The product of two consecutive positive multiples of four is 192. What is the sum of the two numbers?
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Let 
= the first positive number and 
= the second positive number
The equation to solve becomes 
We solve this quadratic equation by multiplying it out and setting it equal to 0. The next step is to factor.
Let
The equation to solve becomes
We solve this quadratic equation by multiplying it out and setting it equal to 0. The next step is to factor.
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Two consecutive positive multiples of three have a product of 504. What is the sum of the two numbers?
Two consecutive positive multiples of three have a product of 504. What is the sum of the two numbers?
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Let 
= the first positive number and 
= the second positive number.
So the equation to solve is 
We multiply out the equation and set it equal to zero before factoring.
thus the two numbers are 21 and 24 for a sum of 45.
Let
So the equation to solve is
We multiply out the equation and set it equal to zero before factoring.
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Two consecutive positive numbers have a product of 420. What is the sum of the two numbers?
Two consecutive positive numbers have a product of 420. What is the sum of the two numbers?
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Let 
= first positive number and 
= second positive number
So the equation to solve becomes 
Using the distributive property, multiply out the equation and then set it equal to 0. Next factor to solve the quadratic.
Let
So the equation to solve becomes
Using the distributive property, multiply out the equation and then set it equal to 0. Next factor to solve the quadratic.
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A rectangle has a perimeter of 
and an area of 
What is the difference between the length and width?
A rectangle has a perimeter of
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For a rectangle, 
and 
where 
= width and 
= length.
So we get two equations with two unknowns:
Making a substitution we get
Solving the quadratic equation we get 
or 
.
The difference is 
.
For a rectangle,
So we get two equations with two unknowns:
Making a substitution we get
Solving the quadratic equation we get
The difference is
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Let f(x) = 2_x_2 – 4_x_ + 1 and g(x) = (_x_2 + 16)(1/2). If k is a negative number such that f(k) = 31, then what is the value of (f(g(k))?
Let f(x) = 2_x_2 – 4_x_ + 1 and g(x) = (_x_2 + 16)(1/2). If k is a negative number such that f(k) = 31, then what is the value of (f(g(k))?
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In order to find the value of f(g(k)), we will first need to find k. We are told that f(k) = 31, so we can write an expression for f(k) and solve for k.
f(x) = 2_x_2 – 4_x_ + 1
f(k) = 2_k_2 – 4_k_ + 1 = 31
Subtract 31 from both sides.
2_k_2 – 4_k –_ 30 = 0
Divide both sides by 2.
k_2 – 2_k – 15 = 0
Now, we can factor this by thinking of two numbers that multiply to give –15 and add to give –2. These two numbers are –5 and 3.
k_2 –2_k – 15 = (k – 5)(k + 3) = 0
We can set each factor equal to 0 to find the values for k.
k – 5 = 0
Add 5 to both sides.
k = 5
Now we set k + 3 = 0.
Subtract 3 from both sides.
k = –3
This means that k could be either 5 or –3. However, we are told that k is a negative number, which means k = –3.
Finally, we can evaluate the expression f(g(–3)). First we need to find g(–3).
g(x) = (_x_2 + 16)(1/2)
g(–3) = ((–3)2 + 16)(1/2)
= (9 + 16)(1/2)
= 25(1/2)
Raising something to the one-half power is the same as taking the square root.
25(1/2) = 5
Now that we know g(–3) = 5, we must find f(5).
f(5) = 2(5)2 – 4(5) + 1
= 2(25) – 20 + 1 = 31
The answer is 31.
In order to find the value of f(g(k)), we will first need to find k. We are told that f(k) = 31, so we can write an expression for f(k) and solve for k.
f(x) = 2_x_2 – 4_x_ + 1
f(k) = 2_k_2 – 4_k_ + 1 = 31
Subtract 31 from both sides.
2_k_2 – 4_k –_ 30 = 0
Divide both sides by 2.
k_2 – 2_k – 15 = 0
Now, we can factor this by thinking of two numbers that multiply to give –15 and add to give –2. These two numbers are –5 and 3.
k_2 –2_k – 15 = (k – 5)(k + 3) = 0
We can set each factor equal to 0 to find the values for k.
k – 5 = 0
Add 5 to both sides.
k = 5
Now we set k + 3 = 0.
Subtract 3 from both sides.
k = –3
This means that k could be either 5 or –3. However, we are told that k is a negative number, which means k = –3.
Finally, we can evaluate the expression f(g(–3)). First we need to find g(–3).
g(x) = (_x_2 + 16)(1/2)
g(–3) = ((–3)2 + 16)(1/2)
= (9 + 16)(1/2)
= 25(1/2)
Raising something to the one-half power is the same as taking the square root.
25(1/2) = 5
Now that we know g(–3) = 5, we must find f(5).
f(5) = 2(5)2 – 4(5) + 1
= 2(25) – 20 + 1 = 31
The answer is 31.
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I. real
II. rational
III. distinct
Which of the descriptions characterizes the solutions of the equation 2x2 – 6x + 3 = 0?
I. real
II. rational
III. distinct
Which of the descriptions characterizes the solutions of the equation 2x2 – 6x + 3 = 0?
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The equation in the problem is quadratic, so we can use the quadratic formula to solve it. If an equation is in the form _ax_2 + bx + c = 0, where a, b, and c are constants, then the quadratic formula, given below, gives us the solutions of x.
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In this particular problem, a = 2, b = –6, and c = 3.
The value under the square-root, b_2 – 4_ac, is called the discriminant, and it gives us important information about the nature of the solutions of a quadratic equation.
If the discriminant is less than zero, then the roots are not real, because we would be forced to take the square root of a negative number, which yields an imaginary result. The discriminant of the equation we are given is (–6)2 – 4(2)(3) = 36 – 24 = 12 > 0. Because the discriminant is not negative, the solutions to the equation will be real. Thus, option I is correct.
The discriminant can also tell us whether the solutions of an equation are rational or not. If we take the square root of the discriminant and get a rational number, then the solutions of the equation must be rational. In this problem, we would need to take the square root of 12. However, 12 is not a perfect square, so taking its square root would produce an irrational number. Therefore, the solutions to the equation in the problem cannot be rational. This means that choice II is incorrect.
Lastly, the discriminant tells us if the roots to an equation are distinct (different from one another). If the discriminant is equal to zero, then the solutions of x become (–b + 0)/2_a_ and (–b – 0)/2_a_, because the square root of zero is 0. Notice that (–b + 0)/2_a_ is the same as (–b – 0)/2_a_. Thus, if the discriminant is zero, then the roots of the equation are the same, i.e. indistinct. In this particular problem, the discriminant = 12, which doesn't equal zero. This means that the two roots will be different, i.e. distinct. Therefore, choice III applies.
The answer is choices I and III only.
The equation in the problem is quadratic, so we can use the quadratic formula to solve it. If an equation is in the form _ax_2 + bx + c = 0, where a, b, and c are constants, then the quadratic formula, given below, gives us the solutions of x.
In this particular problem, a = 2, b = –6, and c = 3.
The value under the square-root, b_2 – 4_ac, is called the discriminant, and it gives us important information about the nature of the solutions of a quadratic equation.
If the discriminant is less than zero, then the roots are not real, because we would be forced to take the square root of a negative number, which yields an imaginary result. The discriminant of the equation we are given is (–6)2 – 4(2)(3) = 36 – 24 = 12 > 0. Because the discriminant is not negative, the solutions to the equation will be real. Thus, option I is correct.
The discriminant can also tell us whether the solutions of an equation are rational or not. If we take the square root of the discriminant and get a rational number, then the solutions of the equation must be rational. In this problem, we would need to take the square root of 12. However, 12 is not a perfect square, so taking its square root would produce an irrational number. Therefore, the solutions to the equation in the problem cannot be rational. This means that choice II is incorrect.
Lastly, the discriminant tells us if the roots to an equation are distinct (different from one another). If the discriminant is equal to zero, then the solutions of x become (–b + 0)/2_a_ and (–b – 0)/2_a_, because the square root of zero is 0. Notice that (–b + 0)/2_a_ is the same as (–b – 0)/2_a_. Thus, if the discriminant is zero, then the roots of the equation are the same, i.e. indistinct. In this particular problem, the discriminant = 12, which doesn't equal zero. This means that the two roots will be different, i.e. distinct. Therefore, choice III applies.
The answer is choices I and III only.
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Solve for x.
3_x_2 + 15_x_ – 18 = 0.
Solve for x.
3_x_2 + 15_x_ – 18 = 0.
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First let's see if there is a common term.
3_x_2 + 15_x_ – 18 = 0
We can pull out a 3: 3(x_2 + 5_x – 6) = 0
Divide both sides by 3: x_2 + 5_x – 6 = 0
We need two numbers that sum to 5 and multiply to –6. 6 and –1 work.
(x + 6)(x – 1) = 0
x = –6 or x = 1
First let's see if there is a common term.
3_x_2 + 15_x_ – 18 = 0
We can pull out a 3: 3(x_2 + 5_x – 6) = 0
Divide both sides by 3: x_2 + 5_x – 6 = 0
We need two numbers that sum to 5 and multiply to –6. 6 and –1 work.
(x + 6)(x – 1) = 0
x = –6 or x = 1
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The expression 
is equal to 0 when 
and 
The expression
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Factor the expression and set each factor equal to 0:
Factor the expression and set each factor equal to 0:
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Two positive consecutive multiples of four have a product of 96. What is the sum of the two numbers?
Two positive consecutive multiples of four have a product of 96. What is the sum of the two numbers?
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Let 
= the first number and 
= the second number.
So the equation to solve becomes 
. This quadratic equation needs to be multiplied out and set equal to zero before factoring. Then set each factor equal to zero and solve. Only positive numbers are correct, so the answer is 
.
Let
So the equation to solve becomes
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Two consecutive positive odd numbers have a product of 35. What is the sum of the two numbers?
Two consecutive positive odd numbers have a product of 35. What is the sum of the two numbers?
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Let 
= first positive number and 
= second positive number.
The equation to solve becomes
We multiply out this quadratic equation and set it equal to 0, then factor.
Let
The equation to solve becomes
We multiply out this quadratic equation and set it equal to 0, then factor.
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Find the zeros of the following function.
Find the zeros of the following function.
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To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (2, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (-3, or b in the standard quadratic formula). Because their product is positive (2) and the sum is negative, that must mean that they both have negative signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and 2, as the product of 1 and 2 is 2, and sum of 1 and 2 is 3. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for 
.
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (2, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (-3, or b in the standard quadratic formula). Because their product is positive (2) and the sum is negative, that must mean that they both have negative signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and 2, as the product of 1 and 2 is 2, and sum of 1 and 2 is 3. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
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Find the zeros of the following function.
Find the zeros of the following function.
Tap to reveal answer
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (6, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (5, or b in the standard quadratic formula). Because their product is positive (6) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 2 and 3, as the product of 2 and 3 is 6, and sum of 2 and 3 is 5. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for .
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (6, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (5, or b in the standard quadratic formula). Because their product is positive (6) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 2 and 3, as the product of 2 and 3 is 6, and sum of 2 and 3 is 5. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
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Find all possible zeros of the following function.
Find all possible zeros of the following function.
Tap to reveal answer
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (1, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (2, or b in the standard quadratic formula). Because their product is positive (1) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and 1, as the product of 1 and 1 is 1, and sum of 1 and 1 is 2. So, this results in the expression's factored form looking like...
From here, set the binomial equal to zero and solve for .
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zero of the function is,
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (1, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (2, or b in the standard quadratic formula). Because their product is positive (1) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and 1, as the product of 1 and 1 is 1, and sum of 1 and 1 is 2. So, this results in the expression's factored form looking like...
From here, set the binomial equal to zero and solve for
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zero of the function is,
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Find all possible zeros for the following function.
Find all possible zeros for the following function.
Tap to reveal answer
To find the zeros of the function using factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (4, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (-4, or b in the standard quadratic formula). Because their product is positive (4) and the sum is negative, that must mean that they both have negative signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are -2 and -2, as the product of -2 and -2 is 4, and sum of -2 and -2 is -4. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for . Since the binomials are the same, there will only be one zero.
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zero of the function is,
To find the zeros of the function using factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (4, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (-4, or b in the standard quadratic formula). Because their product is positive (4) and the sum is negative, that must mean that they both have negative signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are -2 and -2, as the product of -2 and -2 is 4, and sum of -2 and -2 is -4. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zero of the function is,
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Find all possible zeros for the following function.
Find all possible zeros for the following function.
Tap to reveal answer
To find the zeros of the function, use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (-1, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (0, or b in the standard quadratic formula). Because their product is negative (-1) and the sum is zero, that must mean that they have different signs but the same absolute value.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and -1, as the product of 1 and -1 is -1, and sum of 1 and -1 is 0. So, this results in the expression's factored form looking like...
This is known as a difference of squares.
From here, set each binomial equal to zero and solve for 
.
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
To find the zeros of the function, use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (-1, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (0, or b in the standard quadratic formula). Because their product is negative (-1) and the sum is zero, that must mean that they have different signs but the same absolute value.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and -1, as the product of 1 and -1 is -1, and sum of 1 and -1 is 0. So, this results in the expression's factored form looking like...
This is known as a difference of squares.
From here, set each binomial equal to zero and solve for
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
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Find all possible zeros for the following function.
Find all possible zeros for the following function.
Tap to reveal answer
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (3, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (4, or b in the standard quadratic formula). Because their product is positive (3) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and 3, as the product of 1 and 3 is 3, and sum of 1 and 3 is 4. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for .
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore, the zeros are,
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (3, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (4, or b in the standard quadratic formula). Because their product is positive (3) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 1 and 3, as the product of 1 and 3 is 3, and sum of 1 and 3 is 4. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore, the zeros are,
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Find all possible zeros for the following function.
Find all possible zeros for the following function.
Tap to reveal answer
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (20, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (9, or b in the standard quadratic formula). Because their product is positive (20) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 4 and 5, as the product of 4 and 5 is 20, and sum of 4 and 5 is 9. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for .
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros are,
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (20, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (9, or b in the standard quadratic formula). Because their product is positive (20) and the sum is positive, that must mean that they both have positive signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 4 and 5, as the product of 4 and 5 is 20, and sum of 4 and 5 is 9. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros are,
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Find all possible zeros for the following function.
Find all possible zeros for the following function.
Tap to reveal answer
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (-4, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (3, or b in the standard quadratic formula). Because their product is negative (-4) and the sum is positive, that must mean that they have opposite signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 4 and -1, as the product of 4 and -1 is -4, and sum of 4 and -1 is 3. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for .
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
To find the zeros of the function use factoring.
Set up the expression in factored form, leaving blanks for the numbers that are not yet known.
At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (-4, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (3, or b in the standard quadratic formula). Because their product is negative (-4) and the sum is positive, that must mean that they have opposite signs.
Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 4 and -1, as the product of 4 and -1 is -4, and sum of 4 and -1 is 3. So, this results in the expression's factored form looking like...
From here, set each binomial equal to zero and solve for
and
To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.
Therefore the zeros of the function are,
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