Answer :

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]