How to factor the quadratic equation - GRE Quantitative Reasoning
Card 1 of 80
Solve for x: x2 = 45 - 12x
Solve for x: x2 = 45 - 12x
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For quadratic equations, you need get the equation into the form:
Ax2 + Bx + C = 0. Therefore subtract 45 and add 12 x to both sides of x2 = 45 - 12x. This will give you:
x2 +12x - 45 = 0
Following this, you must factor the quadratic element. Since C is negative, your groups will have to be positive and negative. We know that 45 factors into: 1 * 45, 5 * 9, and 15 * 3. The correct pair is 15 and 3.
(x + 15) (x - 3) = 0
Now, we set each of these factors equal to 0 since we know that if either is 0, the whole equation will be zero (hence solving for the correct value of x):
x + 15 = 0; x = -15
x - 3 = 0; x = 3
Therefore x is -15 or 3.
For quadratic equations, you need get the equation into the form:
Ax2 + Bx + C = 0. Therefore subtract 45 and add 12 x to both sides of x2 = 45 - 12x. This will give you:
x2 +12x - 45 = 0
Following this, you must factor the quadratic element. Since C is negative, your groups will have to be positive and negative. We know that 45 factors into: 1 * 45, 5 * 9, and 15 * 3. The correct pair is 15 and 3.
(x + 15) (x - 3) = 0
Now, we set each of these factors equal to 0 since we know that if either is 0, the whole equation will be zero (hence solving for the correct value of x):
x + 15 = 0; x = -15
x - 3 = 0; x = 3
Therefore x is -15 or 3.
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36x2 -12x - 15 = 0
Solve for x
36x2 -12x - 15 = 0
Solve for x
Tap to reveal answer
36x2 - 12x - 15 = 0
Factor the equation:
(6x + 3)(6x - 5) = 0
Set each side equal to zero
6x + 3 = 0
x = -3/6 = -1/2
6x – 5 = 0
x = 5/6
36x2 - 12x - 15 = 0
Factor the equation:
(6x + 3)(6x - 5) = 0
Set each side equal to zero
6x + 3 = 0
x = -3/6 = -1/2
6x – 5 = 0
x = 5/6
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64x2 + 24x - 10 = 0
Solve for x
64x2 + 24x - 10 = 0
Solve for x
Tap to reveal answer
64x2 + 24x - 10 = 0
Factor the equation:
(8x + 5)(8x – 2) = 0
Set each side equal to zero
(8x + 5) = 0
x = -5/8
(8x – 2) = 0
x = 2/8 = 1/4
64x2 + 24x - 10 = 0
Factor the equation:
(8x + 5)(8x – 2) = 0
Set each side equal to zero
(8x + 5) = 0
x = -5/8
(8x – 2) = 0
x = 2/8 = 1/4
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All of the following functions have a exactly one root EXCEPT:
All of the following functions have a exactly one root EXCEPT:
Tap to reveal answer
The roots of an equation are the points at which the function equals zero. We can set each function equal to zero and determine which functions have one root, and which does not.
Another piece of information will help. If a quadratic function has one root, then it must be a perfect square. This is because a quadratic function that is a perfect square can be written in the form (x – a)2. If we set (x – a)2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. Additionally, a quadratic equation is a perfect square if it can be written in the form a2x2 + 2abx + b2 = (ax + b)2.
Let's examine the choice f(x) = 4x2 – 4x+1. To find the roots, we set f(x) = 0.
4x2 – 4x+1 = 0
We notice that 4x2 - 4x + 1 is a perfect square, since we could write it as (2x – 1)2. Thus, this equation has only one root, and it can't be the answer.
If we look at f(x) = x2 –2x + 1, we see that x2 – 2x + 1 is also a perfect square, because it could be written as (x – 1)2. This function also has a single root.
Next, we examine f(x) = (1/4)x2 + x + 1. Let us set f(x) = (1/4)x2 + x + 1 = 0.
(1/4)x2 + x + 1 = 0
We can multiply both sides by four to get rid of the fraction.
x2 + 4x + 4 = 0
(x + 2)2 = 0
This function is also a perfect square and has a single root.
Now consider the choice f(x) = (–1/9)x2 + 6x – 81.
f(x) = (–1/9)x2 + 6x – 81 = 0
Multiply both sides by –9.
x2 – 54x + 729 = 0
(x – 27)2 = 0.
Finally, let's look at f(x) = 9x2 – 6x + 4. This CANNOT be written as a perfect square, because it is not in the form a2x2 + 2abx + b2 = (ax + b)2. It might be tempting to think that 9x2 - 6x + 4 = (3x - 2)2, but it does NOT, because (3x – 2)2 = 9x2 – 12x + 4. Therefore, because 9x2 – 6x + 4 is not a perfect square, it doesn't have exactly one root.
The roots of an equation are the points at which the function equals zero. We can set each function equal to zero and determine which functions have one root, and which does not.
Another piece of information will help. If a quadratic function has one root, then it must be a perfect square. This is because a quadratic function that is a perfect square can be written in the form (x – a)2. If we set (x – a)2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. Additionally, a quadratic equation is a perfect square if it can be written in the form a2x2 + 2abx + b2 = (ax + b)2.
Let's examine the choice f(x) = 4x2 – 4x+1. To find the roots, we set f(x) = 0.
4x2 – 4x+1 = 0
We notice that 4x2 - 4x + 1 is a perfect square, since we could write it as (2x – 1)2. Thus, this equation has only one root, and it can't be the answer.
If we look at f(x) = x2 –2x + 1, we see that x2 – 2x + 1 is also a perfect square, because it could be written as (x – 1)2. This function also has a single root.
Next, we examine f(x) = (1/4)x2 + x + 1. Let us set f(x) = (1/4)x2 + x + 1 = 0.
(1/4)x2 + x + 1 = 0
We can multiply both sides by four to get rid of the fraction.
x2 + 4x + 4 = 0
(x + 2)2 = 0
This function is also a perfect square and has a single root.
Now consider the choice f(x) = (–1/9)x2 + 6x – 81.
f(x) = (–1/9)x2 + 6x – 81 = 0
Multiply both sides by –9.
x2 – 54x + 729 = 0
(x – 27)2 = 0.
Finally, let's look at f(x) = 9x2 – 6x + 4. This CANNOT be written as a perfect square, because it is not in the form a2x2 + 2abx + b2 = (ax + b)2. It might be tempting to think that 9x2 - 6x + 4 = (3x - 2)2, but it does NOT, because (3x – 2)2 = 9x2 – 12x + 4. Therefore, because 9x2 – 6x + 4 is not a perfect square, it doesn't have exactly one root.
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Find the roots of the equation 
.
Find the roots of the equation
Tap to reveal answer
To factor this, we need to find a pair of numbers that multiplies to 6 and sums to 5. The numbers 2 and 3 work. (2 * 3 = 6 and 2 + 3 = 5)
(x + 2)(x + 3) = 0
x = –2 or x = –3
To factor this, we need to find a pair of numbers that multiplies to 6 and sums to 5. The numbers 2 and 3 work. (2 * 3 = 6 and 2 + 3 = 5)
(x + 2)(x + 3) = 0
x = –2 or x = –3
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Which of the following is a root of the function 
?
Which of the following is a root of the function
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The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the 
is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x = 
.
The answer is therefore 
.
The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x =
The answer is therefore
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Given 
and 
, find the value of 
.
Given
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We can factor the quadratic equation into 
.
Then we can see that 
.
Therefore, 
becomes 
and 
.
We can factor the quadratic equation into
Then we can see that
Therefore,
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In the equation 
what are the values of 
?
In the equation 
Tap to reveal answer
To solve, begin by factoring the equation. We know that in our two factors, one will begin with 
and one will begin with 
. Fiddling with different factors of 
(we are looking for two numbers that, when multiplied by 
and 
separately, will add to 
), we come to the following:
(If you are unsure, double check by expanding the equation to match the original)
Now, set the each factor equal to 0:
Do the same for the second factor:
Therefore, our two values of 
are 
and 
.
Alternatively, this problem can be solved by plugging each answer choice into the original equation and finding which set of numbers make the equation equate to 0.
To solve, begin by factoring the equation. We know that in our two factors, one will begin with
(If you are unsure, double check by expanding the equation to match the original)
Now, set the each factor equal to 0:
Do the same for the second factor:
Therefore, our two values of
Alternatively, this problem can be solved by plugging each answer choice into the original equation and finding which set of numbers make the equation equate to 0.
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Solve for x: x2 = 45 - 12x
Solve for x: x2 = 45 - 12x
Tap to reveal answer
For quadratic equations, you need get the equation into the form:
Ax2 + Bx + C = 0. Therefore subtract 45 and add 12 x to both sides of x2 = 45 - 12x. This will give you:
x2 +12x - 45 = 0
Following this, you must factor the quadratic element. Since C is negative, your groups will have to be positive and negative. We know that 45 factors into: 1 * 45, 5 * 9, and 15 * 3. The correct pair is 15 and 3.
(x + 15) (x - 3) = 0
Now, we set each of these factors equal to 0 since we know that if either is 0, the whole equation will be zero (hence solving for the correct value of x):
x + 15 = 0; x = -15
x - 3 = 0; x = 3
Therefore x is -15 or 3.
For quadratic equations, you need get the equation into the form:
Ax2 + Bx + C = 0. Therefore subtract 45 and add 12 x to both sides of x2 = 45 - 12x. This will give you:
x2 +12x - 45 = 0
Following this, you must factor the quadratic element. Since C is negative, your groups will have to be positive and negative. We know that 45 factors into: 1 * 45, 5 * 9, and 15 * 3. The correct pair is 15 and 3.
(x + 15) (x - 3) = 0
Now, we set each of these factors equal to 0 since we know that if either is 0, the whole equation will be zero (hence solving for the correct value of x):
x + 15 = 0; x = -15
x - 3 = 0; x = 3
Therefore x is -15 or 3.
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36x2 -12x - 15 = 0
Solve for x
36x2 -12x - 15 = 0
Solve for x
Tap to reveal answer
36x2 - 12x - 15 = 0
Factor the equation:
(6x + 3)(6x - 5) = 0
Set each side equal to zero
6x + 3 = 0
x = -3/6 = -1/2
6x – 5 = 0
x = 5/6
36x2 - 12x - 15 = 0
Factor the equation:
(6x + 3)(6x - 5) = 0
Set each side equal to zero
6x + 3 = 0
x = -3/6 = -1/2
6x – 5 = 0
x = 5/6
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64x2 + 24x - 10 = 0
Solve for x
64x2 + 24x - 10 = 0
Solve for x
Tap to reveal answer
64x2 + 24x - 10 = 0
Factor the equation:
(8x + 5)(8x – 2) = 0
Set each side equal to zero
(8x + 5) = 0
x = -5/8
(8x – 2) = 0
x = 2/8 = 1/4
64x2 + 24x - 10 = 0
Factor the equation:
(8x + 5)(8x – 2) = 0
Set each side equal to zero
(8x + 5) = 0
x = -5/8
(8x – 2) = 0
x = 2/8 = 1/4
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All of the following functions have a exactly one root EXCEPT:
All of the following functions have a exactly one root EXCEPT:
Tap to reveal answer
The roots of an equation are the points at which the function equals zero. We can set each function equal to zero and determine which functions have one root, and which does not.
Another piece of information will help. If a quadratic function has one root, then it must be a perfect square. This is because a quadratic function that is a perfect square can be written in the form (x – a)2. If we set (x – a)2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. Additionally, a quadratic equation is a perfect square if it can be written in the form a2x2 + 2abx + b2 = (ax + b)2.
Let's examine the choice f(x) = 4x2 – 4x+1. To find the roots, we set f(x) = 0.
4x2 – 4x+1 = 0
We notice that 4x2 - 4x + 1 is a perfect square, since we could write it as (2x – 1)2. Thus, this equation has only one root, and it can't be the answer.
If we look at f(x) = x2 –2x + 1, we see that x2 – 2x + 1 is also a perfect square, because it could be written as (x – 1)2. This function also has a single root.
Next, we examine f(x) = (1/4)x2 + x + 1. Let us set f(x) = (1/4)x2 + x + 1 = 0.
(1/4)x2 + x + 1 = 0
We can multiply both sides by four to get rid of the fraction.
x2 + 4x + 4 = 0
(x + 2)2 = 0
This function is also a perfect square and has a single root.
Now consider the choice f(x) = (–1/9)x2 + 6x – 81.
f(x) = (–1/9)x2 + 6x – 81 = 0
Multiply both sides by –9.
x2 – 54x + 729 = 0
(x – 27)2 = 0.
Finally, let's look at f(x) = 9x2 – 6x + 4. This CANNOT be written as a perfect square, because it is not in the form a2x2 + 2abx + b2 = (ax + b)2. It might be tempting to think that 9x2 - 6x + 4 = (3x - 2)2, but it does NOT, because (3x – 2)2 = 9x2 – 12x + 4. Therefore, because 9x2 – 6x + 4 is not a perfect square, it doesn't have exactly one root.
The roots of an equation are the points at which the function equals zero. We can set each function equal to zero and determine which functions have one root, and which does not.
Another piece of information will help. If a quadratic function has one root, then it must be a perfect square. This is because a quadratic function that is a perfect square can be written in the form (x – a)2. If we set (x – a)2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. Additionally, a quadratic equation is a perfect square if it can be written in the form a2x2 + 2abx + b2 = (ax + b)2.
Let's examine the choice f(x) = 4x2 – 4x+1. To find the roots, we set f(x) = 0.
4x2 – 4x+1 = 0
We notice that 4x2 - 4x + 1 is a perfect square, since we could write it as (2x – 1)2. Thus, this equation has only one root, and it can't be the answer.
If we look at f(x) = x2 –2x + 1, we see that x2 – 2x + 1 is also a perfect square, because it could be written as (x – 1)2. This function also has a single root.
Next, we examine f(x) = (1/4)x2 + x + 1. Let us set f(x) = (1/4)x2 + x + 1 = 0.
(1/4)x2 + x + 1 = 0
We can multiply both sides by four to get rid of the fraction.
x2 + 4x + 4 = 0
(x + 2)2 = 0
This function is also a perfect square and has a single root.
Now consider the choice f(x) = (–1/9)x2 + 6x – 81.
f(x) = (–1/9)x2 + 6x – 81 = 0
Multiply both sides by –9.
x2 – 54x + 729 = 0
(x – 27)2 = 0.
Finally, let's look at f(x) = 9x2 – 6x + 4. This CANNOT be written as a perfect square, because it is not in the form a2x2 + 2abx + b2 = (ax + b)2. It might be tempting to think that 9x2 - 6x + 4 = (3x - 2)2, but it does NOT, because (3x – 2)2 = 9x2 – 12x + 4. Therefore, because 9x2 – 6x + 4 is not a perfect square, it doesn't have exactly one root.
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Find the roots of the equation .
Find the roots of the equation
Tap to reveal answer
To factor this, we need to find a pair of numbers that multiplies to 6 and sums to 5. The numbers 2 and 3 work. (2 * 3 = 6 and 2 + 3 = 5)
(x + 2)(x + 3) = 0
x = –2 or x = –3
To factor this, we need to find a pair of numbers that multiplies to 6 and sums to 5. The numbers 2 and 3 work. (2 * 3 = 6 and 2 + 3 = 5)
(x + 2)(x + 3) = 0
x = –2 or x = –3
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Which of the following is a root of the function ?
Which of the following is a root of the function
Tap to reveal answer
The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the is not equal to 1, we need to multiply this coefficient by the constant, which is –4. When we mutiply 2 and –4, we get –8. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We will now rewrite –7x as –8x + x.
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x = .
The answer is therefore .
The roots of a function are the x intercepts of the function. Whenever a function passes through a point on the x-axis, the value of the function is zero. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x.
This is a quadratic trinomial. Let's see if we can factor it. (We could use the quadratic formula, but it's easier to factor when we can.)
Because the coefficient in front of the
We will then group the first two terms and the last two terms.
We will next factor out a 2_x_ from the first two terms.
Thus, when factored, the original equation becomes (2_x_ + 1)(x – 4) = 0.
We now set each factor equal to zero and solve for x.
Subtract 1 from both sides.
2_x_ = –1
Divide both sides by 2.
Now, we set x – 4 equal to 0.
x – 4 = 0
Add 4 to both sides.
x = 4
The roots of f(x) occur where x =
The answer is therefore
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Given and , find the value of .
Given
Tap to reveal answer
We can factor the quadratic equation into .
Then we can see that .
Therefore, becomes and .
We can factor the quadratic equation into
Then we can see that
Therefore,
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In the equation what are the values of ?
In the equation
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To solve, begin by factoring the equation. We know that in our two factors, one will begin with and one will begin with . Fiddling with different factors of (we are looking for two numbers that, when multiplied by and separately, will add to ), we come to the following:
(If you are unsure, double check by expanding the equation to match the original)
Now, set the each factor equal to 0:
Do the same for the second factor:
Therefore, our two values of are and .
Alternatively, this problem can be solved by plugging each answer choice into the original equation and finding which set of numbers make the equation equate to 0.
To solve, begin by factoring the equation. We know that in our two factors, one will begin with
(If you are unsure, double check by expanding the equation to match the original)
Now, set the each factor equal to 0:
Do the same for the second factor:
Therefore, our two values of
Alternatively, this problem can be solved by plugging each answer choice into the original equation and finding which set of numbers make the equation equate to 0.
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Solve for x: x2 = 45 - 12x
Solve for x: x2 = 45 - 12x
Tap to reveal answer
For quadratic equations, you need get the equation into the form:
Ax2 + Bx + C = 0. Therefore subtract 45 and add 12 x to both sides of x2 = 45 - 12x. This will give you:
x2 +12x - 45 = 0
Following this, you must factor the quadratic element. Since C is negative, your groups will have to be positive and negative. We know that 45 factors into: 1 * 45, 5 * 9, and 15 * 3. The correct pair is 15 and 3.
(x + 15) (x - 3) = 0
Now, we set each of these factors equal to 0 since we know that if either is 0, the whole equation will be zero (hence solving for the correct value of x):
x + 15 = 0; x = -15
x - 3 = 0; x = 3
Therefore x is -15 or 3.
For quadratic equations, you need get the equation into the form:
Ax2 + Bx + C = 0. Therefore subtract 45 and add 12 x to both sides of x2 = 45 - 12x. This will give you:
x2 +12x - 45 = 0
Following this, you must factor the quadratic element. Since C is negative, your groups will have to be positive and negative. We know that 45 factors into: 1 * 45, 5 * 9, and 15 * 3. The correct pair is 15 and 3.
(x + 15) (x - 3) = 0
Now, we set each of these factors equal to 0 since we know that if either is 0, the whole equation will be zero (hence solving for the correct value of x):
x + 15 = 0; x = -15
x - 3 = 0; x = 3
Therefore x is -15 or 3.
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36x2 -12x - 15 = 0
Solve for x
36x2 -12x - 15 = 0
Solve for x
Tap to reveal answer
36x2 - 12x - 15 = 0
Factor the equation:
(6x + 3)(6x - 5) = 0
Set each side equal to zero
6x + 3 = 0
x = -3/6 = -1/2
6x – 5 = 0
x = 5/6
36x2 - 12x - 15 = 0
Factor the equation:
(6x + 3)(6x - 5) = 0
Set each side equal to zero
6x + 3 = 0
x = -3/6 = -1/2
6x – 5 = 0
x = 5/6
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64x2 + 24x - 10 = 0
Solve for x
64x2 + 24x - 10 = 0
Solve for x
Tap to reveal answer
64x2 + 24x - 10 = 0
Factor the equation:
(8x + 5)(8x – 2) = 0
Set each side equal to zero
(8x + 5) = 0
x = -5/8
(8x – 2) = 0
x = 2/8 = 1/4
64x2 + 24x - 10 = 0
Factor the equation:
(8x + 5)(8x – 2) = 0
Set each side equal to zero
(8x + 5) = 0
x = -5/8
(8x – 2) = 0
x = 2/8 = 1/4
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All of the following functions have a exactly one root EXCEPT:
All of the following functions have a exactly one root EXCEPT:
Tap to reveal answer
The roots of an equation are the points at which the function equals zero. We can set each function equal to zero and determine which functions have one root, and which does not.
Another piece of information will help. If a quadratic function has one root, then it must be a perfect square. This is because a quadratic function that is a perfect square can be written in the form (x – a)2. If we set (x – a)2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. Additionally, a quadratic equation is a perfect square if it can be written in the form a2x2 + 2abx + b2 = (ax + b)2.
Let's examine the choice f(x) = 4x2 – 4x+1. To find the roots, we set f(x) = 0.
4x2 – 4x+1 = 0
We notice that 4x2 - 4x + 1 is a perfect square, since we could write it as (2x – 1)2. Thus, this equation has only one root, and it can't be the answer.
If we look at f(x) = x2 –2x + 1, we see that x2 – 2x + 1 is also a perfect square, because it could be written as (x – 1)2. This function also has a single root.
Next, we examine f(x) = (1/4)x2 + x + 1. Let us set f(x) = (1/4)x2 + x + 1 = 0.
(1/4)x2 + x + 1 = 0
We can multiply both sides by four to get rid of the fraction.
x2 + 4x + 4 = 0
(x + 2)2 = 0
This function is also a perfect square and has a single root.
Now consider the choice f(x) = (–1/9)x2 + 6x – 81.
f(x) = (–1/9)x2 + 6x – 81 = 0
Multiply both sides by –9.
x2 – 54x + 729 = 0
(x – 27)2 = 0.
Finally, let's look at f(x) = 9x2 – 6x + 4. This CANNOT be written as a perfect square, because it is not in the form a2x2 + 2abx + b2 = (ax + b)2. It might be tempting to think that 9x2 - 6x + 4 = (3x - 2)2, but it does NOT, because (3x – 2)2 = 9x2 – 12x + 4. Therefore, because 9x2 – 6x + 4 is not a perfect square, it doesn't have exactly one root.
The roots of an equation are the points at which the function equals zero. We can set each function equal to zero and determine which functions have one root, and which does not.
Another piece of information will help. If a quadratic function has one root, then it must be a perfect square. This is because a quadratic function that is a perfect square can be written in the form (x – a)2. If we set (x – a)2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. Additionally, a quadratic equation is a perfect square if it can be written in the form a2x2 + 2abx + b2 = (ax + b)2.
Let's examine the choice f(x) = 4x2 – 4x+1. To find the roots, we set f(x) = 0.
4x2 – 4x+1 = 0
We notice that 4x2 - 4x + 1 is a perfect square, since we could write it as (2x – 1)2. Thus, this equation has only one root, and it can't be the answer.
If we look at f(x) = x2 –2x + 1, we see that x2 – 2x + 1 is also a perfect square, because it could be written as (x – 1)2. This function also has a single root.
Next, we examine f(x) = (1/4)x2 + x + 1. Let us set f(x) = (1/4)x2 + x + 1 = 0.
(1/4)x2 + x + 1 = 0
We can multiply both sides by four to get rid of the fraction.
x2 + 4x + 4 = 0
(x + 2)2 = 0
This function is also a perfect square and has a single root.
Now consider the choice f(x) = (–1/9)x2 + 6x – 81.
f(x) = (–1/9)x2 + 6x – 81 = 0
Multiply both sides by –9.
x2 – 54x + 729 = 0
(x – 27)2 = 0.
Finally, let's look at f(x) = 9x2 – 6x + 4. This CANNOT be written as a perfect square, because it is not in the form a2x2 + 2abx + b2 = (ax + b)2. It might be tempting to think that 9x2 - 6x + 4 = (3x - 2)2, but it does NOT, because (3x – 2)2 = 9x2 – 12x + 4. Therefore, because 9x2 – 6x + 4 is not a perfect square, it doesn't have exactly one root.
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