Exercise 9 Page 237

We can solve an equation of the form

x^{2} + b x = c,

where

b

and

c

are real numbers, by completing the square. Notice that we can always rewrite any quadratic equation in this form by isolating the variable terms and dividing by the coefficient of

x^{2}

if needed. Let's see an example.

4 x^{2} + 24 x + 32 = 0

SubEqn

$LHS - 32 = RHS - 32$

4 x^{2} + 24 x = - 32

DivEqn

$LHS / 4 = RHS / 4$

x^{2} + 6 x = - 8

Now we need to ensure that we have an expression that is a perfect square trinomial. Then, we will be able to factor it as the square of a binomial and take the square root on both sides to find the solutions. Let's recall the expansion of the square of a binomial.

(x + a)^{2} = x^{2} + 2 a x + a^{2}