To find the solution to the equation [tex]\((x+6)(x+2) = 60\)[/tex], follow these steps:
1. Expand the left-hand side of the equation:
[tex]\[
(x+6)(x+2)
\][/tex]
When we expand this product, we get:
[tex]\[
x^2 + 2x + 6x + 12 = x^2 + 8x + 12
\][/tex]
2. Rewrite the equation with the expanded form:
[tex]\[
x^2 + 8x + 12 = 60
\][/tex]
3. Set the equation to zero:
Subtract 60 from both sides to set the equation equal to zero:
[tex]\[
x^2 + 8x + 12 - 60 = 0
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 + 8x - 48 = 0
\][/tex]
4. Solve the quadratic equation:
To solve the quadratic equation [tex]\(x^2 + 8x - 48 = 0\)[/tex], we can use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = -48\)[/tex].
5. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = 8^2 - 4 \cdot 1 \cdot (-48) = 64 + 192 = 256
\][/tex]
6. Find the roots using the quadratic formula:
Since [tex]\(\Delta = 256\)[/tex], we calculate:
[tex]\[
x = \frac{-8 \pm \sqrt{256}}{2 \cdot 1} = \frac{-8 \pm 16}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{-8 + 16}{2} = \frac{8}{2} = 4
\][/tex]
[tex]\[
x = \frac{-8 - 16}{2} = \frac{-24}{2} = -12
\][/tex]
7. Check the given choices:
The possible solutions given are [tex]\(x = -6\)[/tex], [tex]\(x = -4\)[/tex], [tex]\(x = 4\)[/tex], and [tex]\(x = 12\)[/tex].
- [tex]\(x = -6\)[/tex] is not among our solutions.
- [tex]\(x = -4\)[/tex] is not among our solutions.
- [tex]\(x = 4\)[/tex] is one of our obtained solutions.
- [tex]\(x = 12\)[/tex] is not among our solutions.
Given that the valid solution from the provided choices is [tex]\(x = 4\)[/tex], this is the correct answer. Therefore, the solution to [tex]\((x+6)(x+2) = 60\)[/tex] from the given choices is:
[tex]\[
x = 4
\][/tex]
