Question

Find the number to add to x2 – 16x to make it a perfect square trinomial. Write that trinomial as the square of a binomial.add 256; (x - 16)add 64; (x - 3)2o add 32; (x - 162O add 16; (x - 8)

Answer 1

Solution

Given the expression below

Let the number be k

$x^2-16x+k$

To make it a perfect square, we apply the perfect square formula

$(x+a)^2$

Equating both equations

$\begin{gathered} x^2-16x+k=(x+a)^2 \\ x^2-16x+k=(x+a)(x+a) \\ x^2-16x+k=x(x+a)+a(x+a) \\ x^2-16x+k=x^2+ax+ax+a^2 \\ x^2-16x+k=x^2+2ax+a^2 \end{gathered}$

Equating the terms

$\begin{gathered} 2ax=-16x \\ \text{Divide both sides by 2x} \\ \frac{2ax}{2x}=\frac{-16x}{2x} \\ a=-8 \end{gathered}$

Where the term added k is

$\begin{gathered} k=a^2 \\ a=-8 \\ k=(-8)^2 \\ k=64 \end{gathered}$

Substituting for k into the expression

$\begin{gathered} x^2-16x+k \\ k=64 \\ x^2-16x+64 \end{gathered}$

Hence, the trinomial is

$x^2-16x+64$

The number to be added is 64

Writing the trinomial as the square of a binomial becomes

$\begin{gathered} x^2-16x+64 \\ =x^2-8x-8x+64 \\ =x(x-8)-8(x-8)_{} \\ =(x-8)(x-8) \\ =(x-8)^2 \end{gathered}$

Hence, the square of the binomial is

$(x-8)^2$

Thus, the number to be added is 64 and the square of the binomial is (x - 8)²