To solve the quadratic equation [tex]\( 0 = x^2 - 14x + 46 \)[/tex] by completing the square, follow these steps:
1. Start with the given equation:
[tex]\[
x^2 - 14x + 46 = 0
\][/tex]
2. Move the constant term to the other side:
[tex]\[
x^2 - 14x = -46
\][/tex]
3. Complete the square:
- To complete the square, add and subtract [tex]\((\frac{b}{2})^2\)[/tex] where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. In this case, [tex]\(b = -14\)[/tex].
- Calculate [tex]\((\frac{-14}{2})^2 = 49\)[/tex].
So, add and subtract 49 on the left side:
[tex]\[
x^2 - 14x + 49 - 49 = -46
\][/tex]
Which simplifies to:
[tex]\[
(x-7)^2 - 49 = -46
\][/tex]
4. Simplify the equation:
[tex]\[
(x - 7)^2 = 3
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 7 = \pm \sqrt{3}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 7 \pm \sqrt{3}
\][/tex]
Therefore, the solutions to the equation are:
[tex]\[
x = 7 + \sqrt{3} \quad \text{and} \quad x = 7 - \sqrt{3}
\][/tex]
Given the answer choices:
A. [tex]\(z = -14 \pm \sqrt{3}\)[/tex]
B. [tex]\(a = -7 \pm \sqrt{3}\)[/tex]
C. [tex]\(x = 7 \pm \sqrt{3}\)[/tex]
D. [tex]\(a = 14 \pm \sqrt{3}\)[/tex]
The correct answer is:
C. [tex]\(x = 7 \pm \sqrt{3}\)[/tex]
