Completing the Square: Method, Formula and Examples

Completing the square is a method used to solve quadratic equations and to rewrite quadratic expressions in a different form. It helps us to find the solutions of the equation and to understand the properties of a quadratic function, such as its vertex.

In this article, we will learn about, Completing the Square Methods, Completing the Square Formula, Completing the Square Examples and others in detail.

Completing the square is a mathematical technique used to solve quadratic equations, transform quadratic expressions, and understand the properties of quadratic functions. It involves rewriting a quadratic equation in the form of a perfect square trinomial, making it easier to solve and analyze.

Why Use Completing the Square?

• Solving Quadratic Equations: It provides a method to find the roots of any quadratic equation.
• Graphing Parabolas: Helps to rewrite the quadratic function in vertex form, making it easier to identify the vertex and the axis of symmetry.
• Understanding Properties: Reveals important characteristics of the quadratic function, such as its minimum or maximum value.

For the given quadratic equation ax² + bx + c = 0 and to solve the quadratic equation using complete the square method follow the steps:

Step 1: Start with the Standard Form

Step 2: Move the Constant Term

Step 3: Divide by a

Step 4: Find the Number to Complete the Square

Step 5: Rewrite as a Perfect Square

Step 6: Solve for x

For example, factorise x² + 2x – 3 = 0 using all the steps added above.

⇒ x² + 2x = 3

⇒ x² + 2x + (1)² = 3 + (1)²

⇒ (x + 1)² = 4

⇒ x + 1 = ± 2

⇒ x = ± 2 – 1

⇒ x = 1, -3

Completing the square formula is a methodology or procedure for finding the roots of specified quadratic equations, such as ax² + bx + c = 0, where a, b, and c are all real values except a.

ax² + bx + c

Formula for completing the square is: ax² + bx + c ⇒ a(x + m)² + n,

Instead of a lengthy step-by-step approach, we can use the following simple formula to build the square. Find the following to complete the square in ax² + bx + c:

• n = c – (b²/4a)

• m = b/2a

Values substituted in ax² + bx + c = a(x + m)² + n. These formulas are geometrically developed.

Lets assume the quadratic equation is as ax² + bx + c = 0. Follow the steps to solve it using the completing the square approach.

Step 1: Form the equation in such a way that c is on the right side.

Step 2 : Divide the entire equation by an if an is not equal to 1, such that the coefficient of x² equals 1.

Step 3 : On both sides, add the square of half of the term-x coefficient, (b/2a)².

Step 4 : Factor the left side of the equation as the binomial term’s square.

Step 5: On both sides, take the square root.

Step 6 : Find the roots by solving for variable x.

Following these steps one can easily solve quadratic equation by completing sqaure method.

Note: Sometime while using completing the square method one might encounter (-1) inside the roots and in that case the roots of the quadratic equation are complex.

Completing the Square Method i applied by following the steps added above. An example for the same is added below:

Take a look at the quadratic equation ax² + bx + c = 0 (a not equal to 0).

By dividing everything by a, we get

x² + (b/a)x + (c/a) = 0

This can alternatively be written as  (b/2a)²   (by adding and subtracting)

[x + (b/2a)]² – (b/2a)² + (c/a) = 0

[x + (b/2a)]² – [(b² – 4ac)/4a²] = 0

[x + (b/2a)]² = [(b² – 4ac)/4a²]

If b2 – 4ac ≥ 0, then taking the square root, we gets

x + (b/2a) = ± √(b² – 4ac)/ 2a

The quadratic formula is obtained by simplifying this further.

For any quadratic equation ax² + bx + c = 0

Rewrite the Quadratic Equation:

• x² + b/ax + (b/2a)² = -c/a + (b/2a)²

Express as a Perfect Square:

• (x + b/2a)² = (b/2a)² – c/a

Solve for x:

• x = -b/2a ± √{(b/2a)² – c/a}

This method simplifies the process of solving quadratic equation.

Read More,

• Quadratic Function
• Quadratic Inequalities
• Solving Cubic Equations

Example 1: Find the roots of the quadratic equation of the x² + 2x – 12 = 0 by using the method of completing the square.

Solution:

Given Quadratic equation is  x² + 2x – 12 = 0

So as comparing the equation along with standard form,

where b =  2, and c = -12

Then (x + b/2)² = -(c – b²/4)

By substituting the values we get 

(x + 2/2)² = -(-12 – (2²/4) )

(x + 1)²  = 12 + 1

(x + 1)² = 13

x + 1 =  ± √13

x + 1 = ± 3.6

So, x + 1 = +3.6 and x+1 = – 3.6

x = 2.6 , -4.6

Therefore roots for the given equation are 2.6, -4.6.

Example 2: Find the roots of the quadratic equation of the 2x² – 4x – 20 = 0 by using the method of completing the square.

Solution:

Given Quadratic equation is 2x² – 4x – 20 = 0

Supplied equation is not in the form to which the method of completing squares is used, i.e. the x2 coefficient is not 1. To make it one, divide the entire equation by  2 .

then x² – 2x – 10 = 0

So as comparing the equation along with standard form,

where b = – 2, and c = -10

then (x + b/2)² = -(c – b²/4)

by substituting the values we get

(x + (-2/2))² = -( -10 – (2²/4))

(x – 1)²  =  11

x – 1 = ± √11

x – 1  = ± 3.3

So,  x – 1 = + 3.3 and x -1 = -3.3

x = 4.3,  -2.3

Therefore roots for the given equation are 4.3, -2.3.

Example 3: Solve Using the completing the square formula for 3x² – 9x – 27 = 0.

Solution:

Given Quadratic equation is 3x² – 9x – 27 = 0.

we can write it as x² – 3x -9 =0 

So as comparing the equation along with standard form,

where b = – 3, and c = -9

then (x + b/2)² = -(c – b²/4)

by substituting the values we get

(x + (-3/2) )² = -( -9 – (3²/4) )

(x – 1.5 )²  = 11.25

x – 1.5 = ± √11.25

x – 1.5  = ± 3.35

So,  x – 1.5 = + 11.25 and x -1 = -11.25

x = 12.75, -10.25

Therefore roots for the given equation are 12.75, -10.25.

Example 4: Find the number that needs be added to x² – 4x to make it a perfect square trinomial using the completing the square formula.

Solution:

Given expression is x²– 4x

As Comparing the given expression along with ax² + bx + c, 

a = 1; b = -4

Term that should be added to make the above expression a perfect square trinomial using the formula is,

(b/2a)² =  (-4/2(1))²       

(b/2a)² = 4

Therefore the number that needs be added to x² – 4x to make it a perfect square trinomial is 4.

Example 5: Find the number that needs be added to x² + 22x to make it a perfect square trinomial using the completing the square formula.

Solution:

Given expression is x² + 22x

As Comparing the given expression along with ax² + bx + c,

a = 1 ; b = 22

The term that should be added to make the above expression a perfect square trinomial using the formula is,

(b/2a)² =  ( 22/2(1) )²       

(b/2a)²  = 121

Therefore the number that needs to be added to x² + 22x to make it a perfect square trinomial is 121.

Q1. Complete the square for the quadratic expression x² + 8x + 15.

Q2. Solve the quadratic equation x² + 4x + 3=0 by completing the square.

Q3. Rewrite the quadratic equation 2x² + 12x + 7=0 in the form of a perfect square trinomial by completing the square.

Q4. Complete the square for the quadratic expression x² – 6x + 11.

Q5. Solve the quadratic equation x² + 10x + 16=0 by completing the square.

What is the Method of Completing the Square?

The Method of Completing the Square is used for converting a quadratic expression of the form ax² + bx + c to the vertex form a(x + m)² + n.

What is the Formula to Complete the Square?

Completing the square is a method used to solve quadratic equations, convert quadratic expressions into a perfect square trinomial, or rewrite the equation in vertex form.

What is the Perfect Square Formula?

The two perfect squares formula in algebra are (a + b)² and (a – b)² and their value is:

• (a + b)² = a² + b² + 2ab
• (a – b)² = a² + b² – 2ab