Completing the Square
2026-02-28 01:34 Diff
https://brightchamps.com/en-us/math/algebra/completing-the-square

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Last updated on December 15, 2025

To understand and solve quadratic equations, we can use the method of completing the square. This method converts a quadratic expression into vertex form. For instance, this technique transforms a quadratic expression of the form ax² + bx + c into the vertex form, which is a(x - h)² + k, where (h, k) represents the vertex of the parabola. Hence, the left-hand side becomes a perfect square trinomial, helping in rewriting it in vertex form. In this article, we will explore completing the square in detail.

What is Completing the Square?

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In algebra, completing the square is a technique used to rewrite a quadratic expression in a form that is a perfect square. For example, a quadratic equation like:

\(ax^2 + bx + c = 0\)

It can be rewritten (by completing the square) in the form:

  \(a(x + p)^2 + q = 0\)

Where p and q are numbers we calculate during the process.  

How to Complete the Square Method?

We can use the process of completing the square to find the roots or zeros of a quadratic equation or quadratic polynomial, and to factorize the equation. When the expression is not possible to factorize, this technique plays an important role.

For instance, \(x^2 + 2x + 3\) cannot be factorized using real numbers, because there are no two numbers that add to 2 and multiply to 3. In such cases, we use completing the square to rewrite it in a new form: 

  \(a (x + m)^2 + n    \)

This form helps us express the quadratic as a perfect square plus a constant.

    Here, by rewriting the expression as \((x + m)\), we complete the square. 

What are the Steps for Completing the Square?

We have to follow certain steps for the method of completing the square. 

Step 1: Express the quadratic equation as \(x^2 + bx + c\). 
Here, 1 must be the coefficient of \(x^2\). If it is not 1, the number will be a common factor and placed outside. 

Step 2: Find the half of the coefficient of \(x\).

Step 3: Find the square of the number (half of the coefficient). 

Step 4: Add and subtract the square within the expression to maintain equality.

Step 5: To complete the square, factorize the polynomial and use the algebraic identity.   

\(x^2 + 2xy + y^2 = (x + y)^2\)

Or

\(x^2 - 2xy + y^2 = (x - y)^2\)

Now, let us take an example for better understanding. 

Complete the square for \(-4x^2 - 8x - 12\).

Step 1: First, we need to find if the coefficient is 1. Here, the coefficient of x2 is not 1, so the number (-4) is placed outside as a common factor.  

\(-4x^2 - 8x - 12 = -4 (x^2 + 2x + 3)\)

However, the coefficient of x2 is now 1. 

Step 2: Next, we need to find half of the coefficient of x. 
Coefficient of x = 2
Half of 2 = 1 

Step 3: Find the square of 1.
\(1^2 = 1\)

Step 4: Add and subtract the square to the x2 term. 

\(-4 (x^2 + 2x + 3) = -4 (x^2 + 2x + 1 - 1 + 3) \)

Step 5: Factorize the polynomial and apply the algebraic identity. 

\(x^2 + 2xy + y^2 = (x + y)^2\)


Here, the first three terms: \(x^2 + 2x + 1  \)

Last two terms: \(-1  + 3 \)

\(x^2 + 2x + 1 = (x + 1)^2 \)

\(-4 (x^2 + 2x + 1 - 1 + 3) = -4 ((x + 1)² - 1 + 3)\)

Step 6: Next, simplify the last two terms: \(-1 + 3 ​​​​​​​\)
\(-1 + 3 = 2 \)

Now, the expression becomes:

\(-4 (( x + 1)^2 + 2) \)

Step 7: Next, we can distribute the -4:

\(-4 (x + 1)^2 - 8 ​​​​​​​\)

Hence, the final result is:

\(-4x^2 - 8x - 12 = -4 (x + 1)^2 - 8\)

Now, this is the completed square form:

\(a (x + m)^2 + n \)

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What is the Formula for Completing the Square?

Converting a quadratic equation or polynomial into a perfect square with an extra constant is known as the process of completing the square. 

For a quadratic expression: \( ax^2 + bx + c = 0 \), where, a, b, and c are real numbers, but a is not equal to 0. 

The formula for completing the square is:

\(ax² + bx + c = a (x + m)² + n \)

Here,

\(m = \frac{b}{2a} \)

\(n = c - (\frac{b^2}{4a}) \)


When completing the square of a given expression, we first need to find the values of m and n, then substitute these values into the formula. 

Now, let us understand the application of the formula for completing the square. 

For example, \(x^2 - 4x - 8 = 0 \)

Step 1: Now, we can apply the formula for completing the square:

\(ax² + bx + c = a (x + m)² + n \)

Step 2: Here,

a = 1
b = -4 
c = -8 

Step 3: Now, we can find the value of m and n.

\(m = \frac{b}{2a} = \frac{-4}{2} (1) = -2\)

\(n = c - (\frac{b^2}{4a}) = -8 - (\frac{(-4)^2}{4} (1))\)

\(n = -8 - (\frac{16}{4}) = -8 - 4 \)

\(n = -12 \)

Step 4: So, the expression becomes: 

\(x^2 - 4x - 8 = (x - 2)^2 - 12 \)

Step 5: Now, we can solve the expression. 
 

\((x - 2)^2 - 12 = 0\)

\((x - 2)^2 = 12 \)

\(x - 2 =  √12  =  2√3 \)

\(x = 2 ∓ 2√3 \)

Therefore, \(x = 2 + 2√3\) or \(x = 2 - 2√3\)

  • For geometric visualization, divide the rectangle. So, b / 2a will be the length of each rectangle.
     
  • Fix one rectangle to the right side of the square and the next one to the bottom of the square. 
  • To complete the geometric square, we need to add a square of area [ (b / 2a)2 ] to x2 + (b / a) x. To retain the value of the expression, we must subtract it. Thus, to complete the square:

    \(x^2 + (\frac{b}{a}) x = x^2 + (\frac{b}{a}) x +(\frac{b}{2a})^2 - (\frac{b}{2a})^2\)

    \(x^2 + (\frac{b}{a}) x = x^2 + (\frac{b}{a}) x + (\frac{b}{2a})2 - \frac{b^2}{4a^2}\)

  • Multiplying and dividing \((\frac{b}{a}) x\) with 2 gives, 

    \(x^2 + (2x × \frac{b}{2a}) + (\frac{b}{2a})^2 - \frac{b^2}{4a^2}\)

  • Now, we can use the identity,

    \(x^2 + 2xy + y^2 = (x + y)^2 \)

    The equation above can be expressed as, 

    \( x^2 + (\frac{b}{a}) x = (x + \frac{b}{2a})^2 - (\frac{b^2}{4a^2})  \)

Now, we had:

\(ax^2 + bx + c = a (x^2 + \frac{b}{ax}) + c\)

  • Next, substitute the completed square form:

     \(a ((x +\frac{b}{2a})^2 - b^2 / 4a^2) + c \)

  • Now, distribute the ‘a’: 

    \(a (x + \frac{b}{2a})^2 - a (\frac{b^2}{4a^2}) + c \)

  • Then, simplify:

    \(a (\frac{b^2}{4a^2}) = \frac{b^2}{4a}  \)

  • Therefore, the expression becomes:

    \(a (x +\frac{ b}{2a})^2 - \frac{b^2}{4a} + c\)

  • Next, arrange the constants: 

    \(a (x + \frac{b}{2a})^2 - (c - \frac{b^2}{4a})\)

    Thus, \(ax^2 + bx + c = a (x + \frac{b}{2a})^2 - (c - \frac{b^2}{4a})\)

  • The expression follows the form:

     \(a (x + m)^2 + n  \)

    Where, \(m = \frac{b}{2a}\) and \(n = c - \frac{b^2}{4a}\)

Derivation of Completing the Square Formula

Step 1: Start with a general quadratic equation

\(​​​​​​​ax² + bx + c = 0,\) where a ≠ 0.

Step 2: Divide the entire equation by a to make the coefficient of x² equal to 1:

\( x^2 + \frac{b}{a}x + \frac{c}{a} = 0\)

Step 3: Move the constant term to the right side:

\( x^2 + \frac{b}{a}x = -\frac{c}{a} \)

Step 4: Take half of the coefficient of x, square it, and add it to both sides.

Half of (b/a) is b/(2a).

Square it: (b/2a)² = b²/(4a²).

Add to both sides:

\( x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2} \)

Step 5: Write the left side as a perfect square:

\( \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)

Step 6: Take square roots on both sides:

\( x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \)

Step 7: Solve for x:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

This gives the quadratic formula.

Tips and Tricks to Master Completing the Squares

Completing the square is a fundamental algebraic technique used to solve quadratic equations, derive the quadratic formula, and analyze parabolas. Here are some useful tips and tricks to master the concept: 
 

  • Ensure the coefficient of \(x^2\) is 1. If it's not, factor it out from the entire equation. This simplifies the process and aligns with standard methods.
  • Divide the linear coefficient by 2 and square it. Take half of the coefficient of 𝑥, square it, and add this value to both sides of the equation. This step creates a perfect square trinomial on the left-hand side.
     
  • They should also remind students to keep the equation balanced by adding and subtracting the same value on both sides.
     
  • Take the square root of both sides, remembering to consider both the positive and negative roots. Then, isolate 𝑥 to find the solutions to the quadratic equation.
     
  • Parents and teachers should help students practice different quadratic equations, even with tricky coefficients or complex solutions, to build confidence in completing the square.

Common Mistakes and How to Avoid Them on Completing the Square

Sometimes, completing the square seems tricky for students when they solve quadratic equations. Thus, they often make some errors that lead them to incorrect answers. Here are some common mistakes and their helpful solutions to prevent them. 

Real-Life Applications of Completing the Square

Understanding the significance of the method of completing the square is useful since it can be applied in various situations. Here are some real-world applications of the concept.

  • In engineering and construction, engineers use the method of completing the square to ensure proper stability and balance. For example, if they need to construct a building that has different shapes, such as squares and rectangles, professionals can use the technique to find the dimensions. 
  • Astronomers and space scientists can use the technique of completing the square to analyze the satellites and orbits of planets.  For instance, to analyze and predict the path of satellites, researchers use the completing the square method. 
  • Companies and organizations can analyze their profit and break-even points using completing the square process. For example, if the profit is \(P(x) = -4x^2 + 24x - 10\), completing squares can determine the number of items needed to maximize profits.
  • In electronics, quadratic equations appear in signal amplitude and voltage optimization. Completing the square helps determine peak values and thresholds.
  • In computer graphics, parabolic curves and quadratic forms are often converted using completing the square to simplify rendering, animation trajectories, or collision detection in games.

Download Worksheets

Problem 1

Complete the square of: x^2 + 8x

Okay, lets begin

\((x + 4)^2 - 16 \)
 

Explanation

Here, the given expression is \(x^2 + 8x\) 
 

  • First, we need to find the half of the coefficient of x: 

    \(​​​​​8 / 2 = 4 \)

  • Then, square it:
    42 = 16 
     
  • Now, add and subtract 16:
    \(x^2 + 8x + 16 - 16  \)
     
  • Here, we take the first three terms:
    \(x^2 + 8x + 16\)
     
  • We want to express it in the form:
    \((x + a)^2 \)
    \((x + a)^2 = x^2 + 2ax + a^2 \)
     
  • In the given expression, \(x^2 + 8x + 16\)
    Coefficient of x is 8. 
     
  • That corresponds to 2a. 
    2a = 8 
    a = 4 
     
  • Therefore, a2 = 42 = 16 
     
  • So, \(x^2 + 8x + 16 = (x + 4)^2\)
     
  • Hence, the expression becomes: 
    \(​​​​​​​(x + 4)^2 - 16  \)

Thus,\( x^2 + 8x = (x + 4)^2 - 16\)

Well explained 👍

Problem 2

Solve the equation, x^2 - 10x

Okay, lets begin

\((x - 5)^2 - 25\) 
 

Explanation

The given expression is \(x^2 - 10x\) 
 

  • Half of -10 = -5 
     
  • Square of -5:  \((-5)^2 = 25\) 
     
  • Next, add and subtract 25:

    \(x^2 - 10x + 25 - 25 \)

  • Rewrite it as:

    \((x - 5)^2 - 25 = (x - 5)^2 \)

  • Hence, the expression becomes:

    \((x - 5)^2 - 25 \)


Thus, \(x^2 - 10x = (x - 5)^2 - 25\)  

Well explained 👍

Problem 3

Solve x^2 + 7x + 5 = 0

Okay, lets begin

\(\frac{(-7 ± √29)}{ 2} \)

Explanation

The given expression is: 
\(x^2 + 7x + 5\)
 

  • Move the constant to the right side.:
    \(x^2 + 7x = - 5\)
     
  • Take the half of the coefficient of x:
    Coefficient of x is 7. 
    Half of 7 is: \(\frac{7}{2}\) 
     
  • Next, square it:

    \(​​​​​​​(\frac{7}{2})^2 = \frac{49}{4} \)

  • So, we need to add 49 / 4 to both sides of the equation.

    \(​​​​​​​x^2 + 7x + \frac{49}{4} = -5 + \frac{49}{4} \)

  • Convert -5 to a fraction with denominator 4: 
    \( -5 = \frac{-20}{4} \)
     
  • Now, add: 
    \(\frac{-20} {4} + \frac{49}{4} = \frac{29}{4}\)
     
  • Hence, the expression is:
    \( x^2 + 7x + \frac{49}{4} = \frac{29}{4}\) 
     
  • Here, the left-hand side is a perfect square trinomial:

    \(x^2 + 7x + \frac{49}{4} = (x + \frac{7}{2})^2 \)

  • Rewrite the equation: 

     \((x + \frac{7}{2})^2 = \frac{29}{4} \)

  • Now, take the square root of both sides:

    \(x + \frac{7}{2} =  √\frac{29} {4} \)

    \(x + \frac{7}{2} =  \frac{√29} {2}\)

  • Next, subtract \(\frac{7}{2}\) from both sides: 

    \(x = - \frac{7}{2} ± \frac{√29}{2} \)

Since the denominators are the same, we can write it as: 

       \(x = \frac{(-7 ± √29)}{2}\)

Well explained 👍

Problem 4

Complete the square of the expression: 2x^2 + 12 x

Okay, lets begin

\(2 (x + 3)^2 - 18  \)

Explanation

  •  First, we need to factor out 2. 
    \(2 (x^2 + 6x) \)
     
  • Half of 6 = 3
     
  • Square it: 32 = 9 
     
  • Add and subtract inside:
    \(2 (x^2 + 6x + 9 - 9) \)
     
  • Rewrite it as:
    \(2 ((x + 3)^2 - 9) = 2 (x + 3)^2 - 18 \)


Thus, \(2x^2 + 12x = 2 (x + 3)^2 - 18  \)

Well explained 👍

Problem 5

Complete the square of the expression: x^2 - 5x

Okay, lets begin

\((\frac{x - 5}{2})^2 - \frac{25}{4}  \)

Explanation

The given expression is:

 \(x^2 - 5x \)

  • Half of \(-5 = \frac{-5}{2}\) 
     
  • Square it: 

    \(​​​​​​​(\frac{-5}{2})^2 = (\frac{25}{4}) \)

  • We add and subtract the square inside the expression, so the value does not change:  

    \(x^2 - 5x = x^2 - 5x + \frac{25}{4} - (\frac{25}{4})\)

  • Now, we apply the identity: 

    \(x^2 - 2xy + y^2 = (x - y)^2 \)

  • Here, \(x^2 - 5x + (\frac{25}{4} )\)is a perfect square trinomial.
     
  • So, it can be written as: 

    \((\frac{x - 5}{2})^2 \)

  • Hence, the expression becomes: 

     \(x^2 - 5x = (\frac{x - 5}{2})^2 - \frac{25}{4} \)

Therefore, the answer is: 

\( x^2 - 5x = (\frac{x - 5}{2})2 - \frac{25}{4}  \)

Well explained 👍

FAQs on Completing the Square

1.What do you mean by completing the square?

In algebra, we use the method of completing the square to transform a quadratic expression, \(ax^2 + bx + c\), into a vertex form \(a (x + m)^2 + n\). It is useful for solving quadratic equations and for finding the minimum or maximum value of a quadratic expression. 

2.Define a perfect square trinomial

It is a quadratic expression like \(a^2 + 2ab + b^2 \) or \(a^2 - 2ab + b^2\) that can be factored as \((a + b)^2\) or \((a - b)^2\). This expression results from squaring a binomial.  

3.Can the coefficient of x2 be a number other than 1?

Yes, the coefficient of x2 can be any number other than 1 in a quadratic equation. If 1 is not the coefficient, divide the entire equation by that coefficient before the process. If 1 is the coefficient of x2, completing the square is easier. 

4.What do we add to complete the square?

To complete the square in an expression \(ax^2 + bx + c\), we need to add and subtract \((\frac{b}{2a})^2\). This converts the quadratic expression into a perfect square trinomial. Thus, it results in \((x + \frac{b}{2a})^2 - (\frac{b}{2a})^2 + c\). 
 

5.What is the purpose of adding and subtracting the same number?

In completing the square, we add and subtract the same number to keep the quadratic equation balanced. So that the value of the equation remains unchanged, without changing the original expression. 
 

6.Why is completing the square important for building my child’s math skills?

It strengthens algebraic thinking, improves problem-solving abilities, and helps students tackle quadratic equations with confidence.

7.Can completing the square be used in subjects other than math?

Yes! It is used in physics for calculating projectile motion, in economics to find maximum profits, and in engineering for designing parabolic structures.

8.How can I help my child practice this topic at home?

Encourage step-by-step practice of different quadratic equations, use visual aids, worksheets, and online interactive videos to reinforce understanding.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.