COMPLETING THE SQUARE METHOD CLASS 10

In this section, you will learn how to solve a quadratic equation by completing the square method.. 

The following steps will be useful to solve a quadratic equation by completing the square. 

Step 1 :

In the given quadratic equation ax² + bx + c = 0, divide the complete equation by a (coefficient of x²). 

If the coefficient of x² is 1 (a = 1), the above process is not required. 

Step 2 :

Move the number term (constant) to the right side of the equation.

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2. 

Examples :

6x should be written as 2(3)(x).

5x should be written as 2(x)(5/2). 

Step 4 :

The result of step 3 will be in the form of 

x² + 2(x)y  =  k

Step 4 :

Now add y² to each side to complete the square on the left side of the equation.  

Then, 

x² + 2(x)y + y²  =  k + y²

Step 5 :

In the result of step 4, if we use the algebraic identity

(a + b)²  =  a² + 2ab + b²

on the left side of the equation, we get 

(x + y)²  =  k + y²

Step 6 :

Solve (x + y)²  =  k + y² for x by taking square root on both sides. 

Solved Examples

Example 1 :

Solve the following quadratic equation by completing the square method.

x² + 6x - 7  =  0

Solution :

Step 1 :

In the quadratic equation x² + 6x - 7 = 0, the coefficient of x² is 1. 

So, we have nothing to do in this step. 

Step 2 :

Add 7 to each side of the equation x² - 6x - 7  =  0.

x² - 6x  =  7

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2. 

Then, 

x² - 6x  =  7

x² - 2(x)(3)  =  7

Step 4 :

Now add 3² to each side to complete the square on the left side of the equation.  

Then, 

x² - 2(x)(3) + 3²  =  7 + 3²

(x - 3)²  =  7 + 9

(x - 3)²  =  16

Take square root on both sides. 

√(x - 3)²  =  √16

x - 3  =  ±4

x - 3  =  -4  or  x - 3  =  4

x  =  -1  or  x  =  7

Example 2 :

Solve the following quadratic equation by completing the square method.

x² + 3x + 1  =  0

Solution :

Step 1 :

In the quadratic equation x² + 3x + 1 = 0, the coefficient of x² is 1. 

So, we have nothing to do in this step. 

Step 2 :

Subtract 1 from each side of the equation x² + 3x + 1 = 0.

x² + 3x  =  -1

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2. 

Then, 

x² + 3x  =  -1

x² + 2(x)(3/2)  =  -1

Step 4 :

Now add (3/2)² to each side to complete the square on the left side of the equation.  

Then, 

x² + 2(x)(3/2) + (3/2)²  =  -1 + (3/2)²

(x + 3/2)²  =  -1 + 9/4

(x + 3/2)²  =  -4/4 + 9/4

(x + 3/2)²  =  (-4 + 9)/4

(x + 3/2)²  =  5/4

Take square root on both sides. 

√(x + 3/2)²  =  √(5/4)

x + 3/2  =  ±√5/2

x + 3/2  =  -√5/2  or  x + 3/2  =  √5/2

x  =  -√5/2 -3/2  or  x  =  √5/2 - 3/2

x  =  (-√5 - 3)/2  or  x  =  (√5 - 3)/2

x  =  -(√5 + 3)/2  or  x  =  (√5 - 3)/2

Example 3 :

Solve the following quadratic equation by completing the square method.

2x² + 5x - 3  =  0

Solution :

Step 1 :

In the given quadratic equation 2x² + 5x - 3 = 0, divide the complete equation by 2 (coefficient of x²). 

  x² + (5/2)x - (3/2)  =  0

Step 2 :

Add 3/2 to each side. 

x² + (5/2)x  =  3/2

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2. 

Then, 

x² + 2(x)(5/4)  =  3/2

Step 4 :

Now add (5/4)² to each side to complete the square on the left side of the equation.  

Then, 

x² + 2(x)(5/4) + (5/4)²  =  3/2 + (5/4)²

(x + 5/4)²  =  3/2 + 25/16

(x + 5/4)²  =  24/16 + 25/16

(x + 5/4)²  =  (24 + 25)/16

(x + 5/4)²  =  49/16

Take square root on both sides. 

√(x + 5/4)²  =  √(49/16)

x + 5/4  =  ± 7/4

x + 5/4  =  -7/4  or  x + 5/4  =  7/4

x  =  -7/4 - 5/4  or  x  =  7/4 - 5/4

x  =  (-7 - 5)/4  or  x  =  (7 - 5)/4

x  =  -12/4  or  x  =  2/4

x  =  -3  or  x  =  1/2

Example 4 :

Solve the following quadratic equation by completing the square method.

4x² + 4bx - (a² - b²)  =  0

Solution :

Solution :

Step 1 :

In the given quadratic equation 4x² + 4bx - (a² - b²)  =  0, divide the complete equation by 4 (coefficient of x²). 

x² + bx - (a² - b²)/4  =  0

Step 2 :

Add (a² - b²)/4 to each side. 

x² + bx  =  (a² - b²)/4

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2. 

Then, 

x² + bx  =  (a² - b²)/4

x² + 2(x)(b/2)  =  (a² - b²)/4

Step 4 :

Now add (b/2)² to each side to complete the square on the left side of the equation.  

Then, 

x² + 2(x)(b/2) + (b/2)²  =  (a² - b²)/4 + (b/2)²

(x + b/2)²  =  (a² - b²)/4 + b²/4

(x + b/2)²  =  (a² - b² + b²)/4

(x + b/2)²  =  a²/4

Take square root on both sides. 

√(x + b/2)²  =  √(a²/4)

x + b/2  =  ± a/2

x + b/2  =  -a/2  or  x + b/2  =  a/2

x  =  -a/2 - b/2  or  x  =  a/2 - b/2

x  =  (-a - b)/2  or  x  =  (a - b)/2

x  =  -(a + b)/2 or  x  =  (a - b)/2

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