To solve the quadratic equation [tex]\( x^2 - 16x + 54 = 0 \)[/tex] by completing the square, we will follow a detailed, step-by-step process. Let's break it down:
1. Start with the equation:
[tex]\[
x^2 - 16x + 54 = 0
\][/tex]
2. Rearrange the equation to isolate the constant term on one side:
[tex]\[
x^2 - 16x = -54
\][/tex]
3. Find the value to complete the square:
To complete the square, take the coefficient of [tex]\( x \)[/tex] (which is -16), divide it by 2, and then square it:
[tex]\[
\left(\frac{-16}{2}\right)^2 = (-8)^2 = 64
\][/tex]
4. Add and subtract this square within the equation:
[tex]\[
x^2 - 16x + 64 - 64 = -54
\][/tex]
Rearrange to make it easier:
[tex]\[
x^2 - 16x + 64 = 64 - 54
\][/tex]
[tex]\[
x^2 - 16x + 64 = 10
\][/tex]
5. Rewrite the left side as a perfect square trinomial:
[tex]\[
(x - 8)^2 = 10
\][/tex]
6. Take the square root of both sides:
[tex]\[
x - 8 = \pm \sqrt{10}
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 8 \pm \sqrt{10}
\][/tex]
Given this method, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] can be identified from the solutions. The solutions in the form of [tex]\( x = a \pm \sqrt{b} \)[/tex] are:
[tex]\[
\begin{array}{l}
x = 8 - \sqrt{10} \\
x = 8 + \sqrt{10}
\end{array}
\][/tex]
Thus, the values are:
[tex]\[
a = 8
\][/tex]
[tex]\[
b = 10
\][/tex]
So, to fill in the box:
[tex]\[
\begin{array}{l}
x = 8 - \sqrt{10} \\
x = 8 + \sqrt{10}
\end{array}
\][/tex]
These are the solutions to the quadratic equation [tex]\( x^2 - 16x + 54 = 0 \)[/tex] by completing the square.
