Question

complete the square to identify the two values that solve the following quadratic equation: $x^2 - 16x - 13 = 4$. enter the two solutions separated by a comma. (1 point)
$x = \square$

Explanation:

Step1: Simplify the equation

First, we simplify the given quadratic equation \(x^{2}-16x - 13=4\) by subtracting 4 from both sides to get it in the form \(x^{2}+bx = c\).
\(x^{2}-16x-13 - 4=4 - 4\)
\(x^{2}-16x-17 = 0\)
Or we can rewrite it as \(x^{2}-16x=17\) (by adding 13 and 4 to both sides: \(x^{2}-16x-13 + 13+4=4 + 13 + 4\) which simplifies to \(x^{2}-16x=17\))

Step2: Complete the square

For a quadratic equation of the form \(x^{2}+bx\), to complete the square, we take half of the coefficient of \(x\), square it, and add it to both sides.
The coefficient of \(x\) is \(- 16\), half of \(-16\) is \(\frac{-16}{2}=-8\), and squaring it gives \((-8)^{2} = 64\).
We add 64 to both sides of the equation \(x^{2}-16x=17\):
\(x^{2}-16x + 64=17 + 64\)

Step3: Rewrite as a perfect square

The left - hand side \(x^{2}-16x + 64\) is a perfect square trinomial, which can be written as \((x - 8)^{2}\).
So we have \((x - 8)^{2}=81\)

Step4: Solve for x

Take the square root of both sides:
\(x-8=\pm\sqrt{81}\)
Since \(\sqrt{81} = 9\), we have two cases:
Case 1: \(x-8 = 9\)
\(x=9 + 8=17\)
Case 2: \(x - 8=-9\)
\(x=-9 + 8=-1\)

Answer:

\(17,-1\)