Equations / Inequalities - PSAT Math
Card 1 of 1561
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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
Tap to reveal answer
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
← Didn't Know|Knew It →
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))?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
Solve for x.
3_x_2 + 15_x_ – 18 = 0.
Solve for x.
3_x_2 + 15_x_ – 18 = 0.
Tap to reveal answer
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
← Didn't Know|Knew It →
The expression 
is equal to 0 when 
and 
The expression
Tap to reveal answer
Factor the expression and set each factor equal to 0:
Factor the expression and set each factor equal to 0:
← Didn't Know|Knew It →
Factor the following equation.
x2 – 16
Factor the following equation.
x2 – 16
Tap to reveal answer
The correct answer is (x + 4)(x – 4)
We neen to factor x2 – 16 to solve. We know that each parenthesis will contain an x to make the x2. We know that the root of 16 is 4 and since it is negative and no value of x is present we can tell that one 4 must be positive and the other negative. If we work it from the multiple choice answers we will see that when multiplying it out we get x2 + 4x – 4x – 16. 4x – 4x cancels out and we are left with our answer.
The correct answer is (x + 4)(x – 4)
We neen to factor x2 – 16 to solve. We know that each parenthesis will contain an x to make the x2. We know that the root of 16 is 4 and since it is negative and no value of x is present we can tell that one 4 must be positive and the other negative. If we work it from the multiple choice answers we will see that when multiplying it out we get x2 + 4x – 4x – 16. 4x – 4x cancels out and we are left with our answer.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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
← Didn't Know|Knew It →
We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Toby's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.
We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Toby's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.
Tap to reveal answer
First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as
where 
represents Joule's age and 
is Newton's age.
The statement, "Newton is Toby's age younger than eleven years" is translated as
where 
is Toby's age.
The third statement, "Toby is one year younger than Joule" is
.
So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get
Plug this equation into the first equation to get
Solve for 
. Add 
to both sides
Divide both sides by 3
So Joules is 9 years old. Plug this value into the third equation to find Toby's age
Toby is 8 years old. Use this value to find Newton's age using the second equation
Now, we have the age of the following dogs:
Joule: 9 years
Newton: 3 years
Toby: 8 years
First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as
where
The statement, "Newton is Toby's age younger than eleven years" is translated as
where
The third statement, "Toby is one year younger than Joule" is
So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get
Plug this equation into the first equation to get
Solve for
Divide both sides by 3
So Joules is 9 years old. Plug this value into the third equation to find Toby's age
Toby is 8 years old. Use this value to find Newton's age using the second equation
Now, we have the age of the following dogs:
Joule: 9 years
Newton: 3 years
Toby: 8 years
← Didn't Know|Knew It →
Teachers at an elementary school have devised a system where a student's good behavior earns him or her tokens. Examples of such behavior include sitting quietly in a seat and completing an assignment on time. Jim sits quietly in his seat 2 times and completes assignments 3 times, earning himself 27 tokens. Jessica sits quietly in her seat 9 times and completes 6 assignments, earning herself 69 tokens. How many tokens is each of these two behaviors worth?
Teachers at an elementary school have devised a system where a student's good behavior earns him or her tokens. Examples of such behavior include sitting quietly in a seat and completing an assignment on time. Jim sits quietly in his seat 2 times and completes assignments 3 times, earning himself 27 tokens. Jessica sits quietly in her seat 9 times and completes 6 assignments, earning herself 69 tokens. How many tokens is each of these two behaviors worth?
Tap to reveal answer
Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly" 
and "completing assignments" 
, then we can easily construct a simple system of equations,
and
.
We can multiply the first equation by 
to yield 
.
This allows us to cancel the 
terms when we add the two equations together. We get 
, or 
.
A quick substitution tells us that 
. So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.
Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly"
and
We can multiply the first equation by
This allows us to cancel the
A quick substitution tells us that
← Didn't Know|Knew It →
Read, but do not solve, the following problem:
Adult tickets to the zoo sell for $11; child tickets sell for $7. One day, 6,035 tickets were sold, resulting in $50,713 being raised. How many adult and child tickets were sold?
If 
and 
stand for the number of adult and child tickets, respectively, which of the following systems of equations can be used to answer this question?
Read, but do not solve, the following problem:
Adult tickets to the zoo sell for $11; child tickets sell for $7. One day, 6,035 tickets were sold, resulting in $50,713 being raised. How many adult and child tickets were sold?
If
Tap to reveal answer
6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets, 
.
Therefore, we can say 
.
The amount of money raised from adult tickets is $11 per ticket mutiplied by 
tickets, or 
dollars; similarly, 
dollars are raised from child tickets. Add these together to get the total amount of money raised:
These two equations form our system of equations.
6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets,
Therefore, we can say
The amount of money raised from adult tickets is $11 per ticket mutiplied by
These two equations form our system of equations.
← Didn't Know|Knew It →
Solve the following story problem:
Jack and Aaron go to the sporting goods store. Jack buys a glove for 
and 
wiffle bats for 
each. Jack has 
left over. Aaron spends all his money on 
hats for 
each and 
jerseys. Aaron started with 
more than Jack. How much does one jersey cost?
Solve the following story problem:
Jack and Aaron go to the sporting goods store. Jack buys a glove for
Tap to reveal answer
Let's call "
" the cost of one jersey (this is the value we want to find)
Let's call the amount of money Jack starts with "
"
Let's call the amount of money Aaron starts with "
"
We know Jack buys a glove for 
and 
bats for 
each, and then has 
left over after. Thus:
simplifying, 
so Jack started with 
We know Aaron buys 
hats for 
each and 
jerseys (unknown cost "
") and spends all his money.
The last important piece of information from the problem is Aaron starts with 
dollars more than Jack. So:
From before we know:
Plugging in:
so Aaron started with 
Finally we plug 
into our original equation for A and solve for x:
Thus one jersey costs 
Let's call "
Let's call the amount of money Jack starts with "
Let's call the amount of money Aaron starts with "
We know Jack buys a glove for
simplifying,
We know Aaron buys
The last important piece of information from the problem is Aaron starts with
From before we know:
Plugging in:
so Aaron started with
Finally we plug
Thus one jersey costs
← Didn't Know|Knew It →
Hannah is selling candles for a school fundraiser all fall. She sets a goal of selling 
candles per month. The number of candles she has remaining for the month can be expressed at the end of each week by the equations 
, where 
is the number of candles and 
is the number of weeks she has sold candles this month. What is the meaning of the value 
in this equation?
Hannah is selling candles for a school fundraiser all fall. She sets a goal of selling 
Tap to reveal answer
Since we know that 
stands for weeks, the answer has to have something to do with the weeks. This eliminates "the number of candles she has remaining for the month." Also, we can eliminate "the number of weeks that she has sold candles this month" because that would be our value for 
, not what we'd multiply 
by. The correct answer is, "the number of candles that she sells each week."
Since we know that 
← Didn't Know|Knew It →
Factor 36_x_2 – 49_y_2.
Factor 36_x_2 – 49_y_2.
Tap to reveal answer
This is a difference of squares. The difference of squares formula is a_2 – b_2 = (a + b)(a – b). In this problem, a = 6_x and b = 7_y.
So 36_x_2 – 49_y_2 = (6_x_ + 7_y_)(6_x_ – 7_y_).
This is a difference of squares. The difference of squares formula is a_2 – b_2 = (a + b)(a – b). In this problem, a = 6_x and b = 7_y.
So 36_x_2 – 49_y_2 = (6_x_ + 7_y_)(6_x_ – 7_y_).
← Didn't Know|Knew It →
If x3 – y3 = 30, and x2 + xy + y2 = 6, then what is x2 – 2xy + y2?
If x3 – y3 = 30, and x2 + xy + y2 = 6, then what is x2 – 2xy + y2?
Tap to reveal answer
First, let's factor x3 – y3 using the formula for difference of cubes.
x3 – y3 = (x – y)(x2 + xy + y2)
We are told that x2 + xy + y2 = 6. Thus, we can substitute 6 into the above equation and solve for x – y.
(x - y)(6) = 30.
Divide both sides by 6.
x – y = 5.
The original questions asks us to find x2 – 2xy + y2. Notice that if we factor x2 – 2xy + y2 using the formula for perfect squares, we obtain the following:
x2 – 2xy + y2 = (x – y)2.
Since we know that (x – y) = 5, (x – y)2 must equal 52, or 25.
Thus, x2 – 2xy + y2 = 25.
The answer is 25.
First, let's factor x3 – y3 using the formula for difference of cubes.
x3 – y3 = (x – y)(x2 + xy + y2)
We are told that x2 + xy + y2 = 6. Thus, we can substitute 6 into the above equation and solve for x – y.
(x - y)(6) = 30.
Divide both sides by 6.
x – y = 5.
The original questions asks us to find x2 – 2xy + y2. Notice that if we factor x2 – 2xy + y2 using the formula for perfect squares, we obtain the following:
x2 – 2xy + y2 = (x – y)2.
Since we know that (x – y) = 5, (x – y)2 must equal 52, or 25.
Thus, x2 – 2xy + y2 = 25.
The answer is 25.
← Didn't Know|Knew It →
if x – y = 4 and x2 – y = 34, what is x?
if x – y = 4 and x2 – y = 34, what is x?
Tap to reveal answer
This can be solved by substitution and factoring.
x2 – y = 34 can be written as y = x2 – 34 and substituted into the other equation: x – y = 4 which leads to x – x2 + 34 = 4 which can be written as x2 – x – 30 = 0.
x2 – x – 30 = 0 can be factored to (x – 6)(x + 5) = 0 so x = 6 and –5 and because only 6 is a possible answer, it is the correct choice.
This can be solved by substitution and factoring.
x2 – y = 34 can be written as y = x2 – 34 and substituted into the other equation: x – y = 4 which leads to x – x2 + 34 = 4 which can be written as x2 – x – 30 = 0.
x2 – x – 30 = 0 can be factored to (x – 6)(x + 5) = 0 so x = 6 and –5 and because only 6 is a possible answer, it is the correct choice.
← Didn't Know|Knew It →