To find the [tex]\(x\)[/tex]-intercepts of the equation [tex]\(x^2 + 6x + 8 = 0\)[/tex] by factoring, follow these steps:
1. Identify the structure of the quadratic equation:
The given quadratic equation is
[tex]\[
x^2 + 6x + 8 = 0
\][/tex]
which is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = 8\)[/tex].
2. Find two numbers that multiply to [tex]\(c = 8\)[/tex] and add up to [tex]\(b = 6\)[/tex]:
We need to find two numbers that multiply to 8 (the constant term) and add up to 6 (the coefficient of [tex]\(x\)[/tex]).
These numbers are 2 and 4, since:
[tex]\[
2 \cdot 4 = 8 \quad \text{and} \quad 2 + 4 = 6
\][/tex]
3. Rewrite the middle term [tex]\(6x\)[/tex] using the two numbers found:
Rewrite [tex]\(6x\)[/tex] as [tex]\(2x + 4x\)[/tex]:
[tex]\[
x^2 + 6x + 8 = x^2 + 2x + 4x + 8
\][/tex]
4. Factor by grouping:
Group the terms to factor by grouping:
[tex]\[
x^2 + 2x + 4x + 8 = (x^2 + 2x) + (4x + 8)
\][/tex]
Factor out the common factors in each group:
[tex]\[
(x(x + 2)) + (4(x + 2))
\][/tex]
Notice that [tex]\(x + 2\)[/tex] is a common factor:
[tex]\[
= (x + 2)(x + 4)
\][/tex]
5. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
Set each factor equal to zero to find the [tex]\(x\)[/tex]-intercepts:
[tex]\[
x + 2 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solve each equation:
[tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
[tex]\[
x + 4 = 0 \implies x = -4
\][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts of the equation [tex]\(x^2 + 6x + 8 = 0\)[/tex] are [tex]\(x = -4\)[/tex] and [tex]\(x = -2\)[/tex].
