**2.4.2 Solving Quadratic Equations – Completing the Square**

**(A) The Perfect Square**

**1.**The expression

*x*

^{2}+ 2

*x*+ 1 can be written in the form (

*x*+ 1)

^{2}, it is called a “perfect square”.

**2.**If the algebraic expression on the left hand side of the quadratic equation is a perfect, the roots can be easily obtained by finding the square roots.

**Example**:

Solve each of the following quadratic equation

(a) (

*x*+ 1)^{2}= 25(b)

*x*^{2}− 8*x*+ 16 = 49

*Solution:*

**(a)**

(

*x*+ 1)^{2}= 25(

*x*+ 1)^{2}= ±√25*x*= −1 ± 5

*x*

**= 5**

**or**

*x***=**

**−6**

**(b)**

*x*

^{2}− 8

*x*+ 16 = 49

(

*x*− 4)^{2}= 49(

*x*− 4) = ±√49*x*= 4 ± 7

*x*

**= 11**

**or**

*x***=**

**−3**

**(B) Solving Quadratic Equation by Completing the Square**

**1.**To solve quadratic equation, we make the left hand side of the equation a perfect square.

**2.**To make any quadratic expression

*x*

^{2}+

*px*into a perfect square, we add the term (

*p*

**/**2)

^{2}to the expression and this will make

**3.**The following shows the steps to solve the equation by using completing the square method for quadratic equation

*ax*

^{2}+

*bx*= –

*c.*

**(a)**Rewrite the equation in the form

*ax*

^{2}+

*bx*= –

*c.*

**(b)**If the coefficient

*a*≠ 1, reduce it to 1 (by dividing).

**(c)**Add (

*p*

**/**2)

^{2}to both sides of the equation.

**(d)**Write the expression on the left hand side as a perfect square.

(e) Solve the equation.