SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE

Solving Quadratic Equations by Completing the Square :

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

To apply completing the square method, the quadratic equation must be in the form of 

ax² + bx + c  =  0

Solving Quadratic Equations by Completing the Square - Steps

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. 

Solving Quadratic Equations by Completing the Square - Examples

Example 1 :

Solve the following quadratic equation by completing the square method.

9x² - 12x + 4  =  0

Solution :

Step 1 :

In the given quadratic equation 9x² - 12x + 4 = 0, divide the complete equation by 9 (coefficient of x²). 

  x² - (12/9)x + (4/9)  =  0

x² - (4/3)x + (4/9)  =  0

Step 2 :

Subtract 4/9 from each side. 

x² - (4/3)x  =  - 4/9

Step 3 :

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

Then, 

x² - (4/3)x  =  - 4/9

x² - 2(x)(2/3)  =  - 4/9

Step 4 :

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

Then, 

x² - 2(x)(2/3) + (2/3)²  =  - 4/9 + (2/3)²

(x - 2/3)²  =  - 4/9 + 4/9

(x - 2/3)²  =  0

Take square root on both sides. 

√(x - 2/3)²  =  √0

x - 2/3  =  0

Add 2/3 to each side. 

x  =  2/3

So, the solution is 2/3. 

Example 2 :

Solve the following quadratic equation by completing the square method.

(5x + 7)/(x - 1)  =  3x + 2

Solution :

Write the given quadratic equation in the form :

ax² + bx + c  =  0

Then, 

(5x + 7)/(x - 1)  =  3x + 2

Multiply each side by (x - 1). 

5x + 7  =  (3x + 2)(x - 1)

Simplify. 

5x + 7  =  3x² - 3x + 2x - 2

5x + 7  =  3x² - x - 2

0  =  3x² - 6x - 9

or

3x² - 6x - 9  =  0

Divide the entire equation by 3.

x² - 2x - 3  =  0

Step 1 :

In the quadratic equation x² - 2x - 3 = 0, the coefficient of x² is 1. 

So, we have nothing to do in this step. 

Step 2 :

Add 3 to each side of the equation x² - 2x - 3 = 0.

x² - 2x  =  3

Step 3 :

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

Then, 

x² - 2x  =  3

x² - 2(x)(1)  =  3

Step 4 :

Now add 1² to each side to complete the square on the left side of the equation.  

Then, 

x² - 2(x)(1) + 1²  =  3 + 1²

(x - 1)²  =  3 + 1

(x - 1)²  =  4

Take square root on both sides. 

√(x - 1)²  =  √4

x - 1  =  ±2

x - 1  =  -2  or  x - 1  =  2

x  =  -1  or  x  =  3

So, the solution is {-1, 3}. 

After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations by completing the square method. 

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