Everything Maths Grade 11

CHAPTER 7. SOLVINGQUADRATIC EQUATIONS 7.3

(x + 1)(x− 11) = 0

∴ x =−1 or x = 11

Example 5: Solving Quadratic Equations by Completing the Square

QUESTION

Solve by completing thesquare:

2 x^2 − 8 x− 16 = 0

SOLUTION

Step 1 : Write the equation in the form ax^2 + bx + c = 0

2 x^2 − 8 x− 16 = 0

Step 2 : Take the constant overto the right hand side of the equation

2 x^2 − 8 x = 16

Step 3 : Check that the coefficient of the x^2 term is 1.

The coefficient of the x^2 term is 2. Therefore, divide bothsides by 2 :

x^2 − 4 x = 8

Step 4 : Take half the coefficient of the x term, square it and addit to both sides

The coefficient of the x term is− 4 ;(− 2 4)=− 2 and (−2)^2 = 4. Therefore:

x^2 − 4 x + 4 = 8 + 4

Step 5 : Write the left hand side as a perfect square

(x− 2)^2 − 12 = 0

Step 6 : Factorise equation as difference of squares

[(x− 2) +

√

12][(x− 2)−

√

12] = 0

Step 7 : Solve for the unknownvalue

[x− 2 +

√

12][x− 2 −

√

12] = 0

∴ x = 2−

√

12 or x = 2 +

√

12

Step 8 : The last three steps canalso be done in a different way