2.2b Solving Quadratic Equations – Completing the Square


2.2b Solving Quadratic Equations – Completing the Square

(A) The Perfect Square
1. The expression x2 + 2x + 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) x2 8x + 16 = 49

Solution:
(a)
(x + 1)2 = 25
(x + 1)2 = ±√25
x = 1 ± 5
x = 5  or  x = 6

(b)
x2 8x + 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 x2 + 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 ax2+ bx = – c.
 (a) Rewrite the equation in the form ax2 + 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.

Click on the image below to learn how to solve quadratic equation by completing the square

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