Exercise 9 Page 237

We can solve an equation of the form $x^2+bx=c,$ 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. 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. By adding the appropriate constant, we can obtain a perfect square trinomial. Let's compare the right-hand side of this formula to the left-hand side of our expression. Notice that the $b$ parameter of our expression corresponds to the factor $2a$ in the expansion for the square of the binomial. We can find the quantity corresponding to $a^2$ that we would have to add from this relationship by solving for $a$ first and then squaring both sides. This means that by adding the term ${\left(\frac{b}{2}\right)}^2$ we can get a perfect square trinomial! Let's try this with the example equation we had at the beginning, $x^2+6x=\text{-}8.$ Notice that here $b=6.$ Then, adding $9$ to both sides of $x^2+6x=\text{-}8$ will let us rewrite the left-hand side as the square of a binomial. As we can see, the solutions for this equation are $x_1=\text{-}3+1=\text{-}2$ and $x_2=\text{-}3-1=\text{-}4.$

Summary of the Procedure

As we saw above, we can complete the square by following the next steps.

1. We isolate the terms containing the variable $x.$
2. We divide both sides by the coefficient of $x^2.$
3. We identify the coefficient of the linear term as $b.$
4. We add ${\left(\frac{b}{2}\right)}^2$ to both sides.