To determine which of the given choices is a solution to the equation [tex]\((x-2)(x+10)=13\)[/tex], follow these steps:
1. Set up the equation:
[tex]\[(x-2)(x+10)= 13\][/tex]
2. Expand the left-hand side of the equation:
When expanding [tex]\((x-2)(x+10)\)[/tex], use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x-2)(x+10) = x^2 + 10x - 2x - 20 = x^2 + 8x - 20
\][/tex]
So, the equation becomes:
[tex]\[
x^2 + 8x - 20 = 13
\][/tex]
3. Move all terms to one side to form a standard quadratic equation:
Subtract 13 from both sides:
[tex]\[
x^2 + 8x - 20 - 13 = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 + 8x - 33 = 0
\][/tex]
4. Solve the quadratic equation [tex]\(x^2 + 8x - 33 = 0\)[/tex]:
This can be done by factoring, completing the square, or using the quadratic formula. We can use the quadratic formula in this case, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = -33\)[/tex].
Plugging in these values:
[tex]\[
x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-33)}}{2 \cdot 1}
\][/tex]
Simplify under the square root:
[tex]\[
x = \frac{-8 \pm \sqrt{64 + 132}}{2}
\][/tex]
[tex]\[
x = \frac{-8 \pm \sqrt{196}}{2}
\][/tex]
[tex]\[
x = \frac{-8 \pm 14}{2}
\][/tex]
This gives two potential solutions:
[tex]\[
x = \frac{-8 + 14}{2} = \frac{6}{2} = 3
\][/tex]
and
[tex]\[
x = \frac{-8 - 14}{2} = \frac{-22}{2} = -11
\][/tex]
5. Check which of the provided choices match the solutions:
The possible choices are:
[tex]\[
x = 3, \quad x = 8, \quad x = 10, \quad x = 11
\][/tex]
From our solutions, we have [tex]\(x = 3\)[/tex] and [tex]\(x = -11\)[/tex]. Therefore, among the provided choices, the correct solution is:
[tex]\[
\boxed{3}
\][/tex]
