Question

Which of the following is equivalent to 0 = 3x2 – 12x – 15 when completing the square?

Answer 1

The first thing we can do is notice that all the factors are divisible by 3. So we can divide both sides by 3 to get:

$\begin{gathered} \frac{3x^2-12x-15}{3}=\frac{0}{3} \\ \frac{3x^2}{3}-\frac{12x}{3}-\frac{15}{3}=0 \\ x^2-4x-5=0 \end{gathered}$

A square can be written in this way:

$(x-a)^2=x^2-2ax+a^2$

Comparing it to our equation, we see the x² is already equal. Comparing the second term, we get:

$\begin{gathered} -4=-2a \\ -2a=-4 \\ a=\frac{-4}{-2} \\ a=2 \end{gathered}$

So, if a = 2, then a² = 4. However, we have -5 instead of 4. To fix this, we can add 9 to both sides:

$\begin{gathered} x^2-4x-5+9=0+9 \\ x^2-4x+4=9 \end{gathered}$

And now we have the right square:

$\begin{gathered} (x-2)^2=x^2-4x+4 \\ so \\ (x-2)^2=9 \end{gathered}$