A completing a square SAT question asks you to rewrite a quadratic expression as a perfect square. This can help you solve an equation, identify an equivalent form, or recognize an important feature of a quadratic.
These questions are usually not difficult because of long calculations. They become difficult when students forget the middle term, use the wrong number to complete the square, or add a value to one side of an equation but not the other.
In this guide, you will learn how to solve a completing a square SAT question step by step, avoid common mistakes, and practice with SAT-style examples.
Completing a Square SAT Question: What It Tests
A completing a square SAT question tests whether you can rewrite a quadratic expression in a cleaner form.
The main idea is to change an expression like this:
x² + bx + c
into a form like this:
(x + a)² + k
or this:
(x − a)² + k
The main skills you need are:
recognizing a quadratic expression;
taking half of the x-coefficient;
squaring that number;
rewriting a trinomial as a perfect square;
keeping both sides of an equation balanced;
checking signs carefully.
For example:
x² + 6x
can become part of a perfect square because half of 6 is 3, and 3² is 9.
So:
x² + 6x + 9 = (x + 3)²
The key number is 9, because it completes the square.
When Should You Complete the Square on the SAT?
You should complete the square on the SAT when factoring is not obvious or when the question asks for an equivalent form of a quadratic expression.
Completing the square is especially useful when:
the quadratic is not easy to factor;
the question asks which expression is equivalent;
the question gives a form like (x − a)² = b;
the question asks for a missing value;
the question involves the vertex form of a quadratic;
the answer choices are written with parentheses and a squared term.
For example, if you see answer choices like these:
A. (x − 4)² − 9 B. (x + 4)² − 9 C. (x − 4)² + 9 D. (x + 4)² + 9
the question is probably testing completing the square or recognizing a completed-square form.
How to Complete the Square on the SAT
To complete the square, focus on the coefficient of x.
For a basic expression like:
x² + bx
use this process:
Step 1: Find the x-Coefficient
The x-coefficient is the number in front of x.
For example, in:
x² + 8x
the x-coefficient is 8.
In:
x² − 10x
the x-coefficient is −10.
Step 2: Take Half of the x-Coefficient
Divide the x-coefficient by 2.
For example:
8 ÷ 2 = 4
and:
−10 ÷ 2 = −5
This number will go inside the parentheses.
Step 3: Square That Number
Now square the number you found.
For example:
4² = 16
and:
(−5)² = 25
This is the number that completes the square.
Step 4: Add the Number Needed to Complete the Square
If you are working with an equation, add the same number to both sides.
For example:
x² + 8x = 5
Half of 8 is 4.
4² = 16.
Add 16 to both sides:
x² + 8x + 16 = 5 + 16
Step 5: Rewrite as a Perfect Square
Now rewrite the left side as a squared binomial:
x² + 8x + 16 = (x + 4)²
So:
x² + 8x = 5
becomes:
(x + 4)² = 21
That is the completed-square form.
Completing a Square SAT Question: Worked Example
Here is a SAT-style completing a square question.
The equation x² − 8x + 7 = 0 can be rewritten in the form:
(x − a)² = b
What is the value of b?
A. 7 B. 9 C. 16 D. 23
Step-by-Step Solution
Start with the equation:
x² − 8x + 7 = 0
Move the constant to the other side:
x² − 8x = −7
Now take half of the x-coefficient.
The x-coefficient is −8.
Half of −8 is −4.
Now square −4:
(−4)² = 16
Add 16 to both sides:
x² − 8x + 16 = −7 + 16
Rewrite the left side as a perfect square:
(x − 4)² = 9
So:
b = 9
Correct Answer
The correct answer is B. 9.
Why This Answer Is Correct
The expression x² − 8x becomes a perfect square when 16 is added:
x² − 8x + 16 = (x − 4)²
Since 16 was added to the left side, it must also be added to the right side:
−7 + 16 = 9
So the completed-square form is:
(x − 4)² = 9
Common Trap
The most common trap is choosing C. 16.
That answer comes from finding the number needed to complete the square, but the question asks for b in the final equation.
The number 16 is added to both sides, but the value of b is:
−7 + 16 = 9
On a completing a square SAT question, always check what the question is asking for before choosing an answer.
Common Mistakes in Completing the Square
Completing the square depends on one precise calculation: take half of the x-coefficient and square it. Most mistakes come from using the wrong coefficient, forgetting a sign, or not keeping the equation balanced.
Mistake 1: Using the Full x-Coefficient
If the expression is:
x² + 10x
do not square 10.
First take half of 10:
10 ÷ 2 = 5
Then square 5:
5² = 25
So:
x² + 10x + 25 = (x + 5)²
Mistake 2: Forgetting to Square the Number
If the expression is:
x² − 6x
half of −6 is −3.
But the number that completes the square is not −3. You must square it:
(−3)² = 9
So:
x² − 6x + 9 = (x − 3)²
Mistake 3: Losing the Sign Inside the Parentheses
The sign inside the parentheses comes from half of the x-coefficient.
For example:
x² − 12x + 36 = (x − 6)²
not:
(x + 6)²
Because:
(x − 6)² = x² − 12x + 36
Mistake 4: Adding to Only One Side of an Equation
If you add a number to complete the square, you must keep the equation balanced.
For example:
x² + 4x = 12
Half of 4 is 2.
2² = 4.
Add 4 to both sides:
x² + 4x + 4 = 12 + 4
So:
(x + 2)² = 16
Do not add 4 only to the left side.
Mistake 5: Forgetting to Factor First When Needed
Sometimes the coefficient of x² is not 1.
For example:
2x² + 12x + 5
Before completing the square, factor 2 from the x² and x terms:
2(x² + 6x) + 5
Now complete the square inside the parentheses.
Half of 6 is 3.
3² = 9.
So:
x² + 6x = (x + 3)² − 9
Then:
2(x² + 6x) + 5 = 2[(x + 3)² − 9] + 5
Distribute:
2(x + 3)² − 18 + 5
So the expression becomes:
2(x + 3)² − 13
This type is more advanced, but it can appear when the SAT asks for an equivalent form.
Completing a Square SAT Practice Questions
Use these SAT-style practice questions to check whether you can complete the square, rewrite quadratic expressions, and avoid common traps.
Question 1
Which expression is equivalent to x² + 6x + 5?
A. (x + 3)² − 4 B. (x + 3)² + 5 C. (x + 6)² − 31 D. (x − 3)² − 4
Answer:
A. (x + 3)² − 4
Explanation:
Start with:
x² + 6x + 5
Half of 6 is 3.
3² = 9.
Rewrite x² + 6x as:
x² + 6x = (x + 3)² − 9
Now include the +5:
x² + 6x + 5 = (x + 3)² − 9 + 5
So:
x² + 6x + 5 = (x + 3)² − 4
The correct answer is A.
Question 2
The equation x² − 12x + 20 = 0 can be rewritten as:
(x − 6)² = k
What is the value of k?
A. 4 B. 16 C. 20 D. 36
Answer:
B. 16
Explanation:
Start with:
x² − 12x + 20 = 0
Move the constant:
x² − 12x = −20
Take half of −12:
−12 ÷ 2 = −6
Square −6:
(−6)² = 36
Add 36 to both sides:
x² − 12x + 36 = −20 + 36
So:
(x − 6)² = 16
The value of k is 16.
Question 3
The expression x² + bx + 25 is a perfect square. If b is positive, what is the value of b?
A. 5 B. 10 C. 20 D. 25
Answer:
B. 10
Explanation:
Since 25 is the last term of the perfect square, think of:
5² = 25
So the expression should match:
(x + 5)²
Expand:
(x + 5)² = x² + 10x + 25
So:
b = 10
The correct answer is B.
Question 4
The expression x² − 4x + 9 can be written in the form:
(x − h)² + k
What is the value of h + k?
A. 3 B. 5 C. 7 D. 9
Answer:
C. 7
Explanation:
Start with:
x² − 4x + 9
Half of −4 is −2.
Square −2:
(−2)² = 4
Rewrite x² − 4x as:
x² − 4x = (x − 2)² − 4
Now add 9:
x² − 4x + 9 = (x − 2)² − 4 + 9
So:
x² − 4x + 9 = (x − 2)² + 5
This means:
h = 2
and:
k = 5
So:
h + k = 2 + 5 = 7
The correct answer is C.
Question 5
Which expression is equivalent to 2x² + 16x + 10?
A. 2(x + 4)² − 22 B. 2(x + 4)² + 10 C. 2(x + 8)² − 118 D. 2(x − 4)² − 22
Answer:
A. 2(x + 4)² − 22
Explanation:
Start with:
2x² + 16x + 10
Factor 2 from the x² and x terms:
2(x² + 8x) + 10
Now complete the square inside the parentheses.
Half of 8 is 4.
4² = 16.
Rewrite:
x² + 8x = (x + 4)² − 16
Substitute this back:
2[(x + 4)² − 16] + 10
Distribute:
2(x + 4)² − 32 + 10
So:
2x² + 16x + 10 = 2(x + 4)² − 22
The correct answer is A.
Use This Skill in a Full SAT Practice Test
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