To solve the quadratic equation [tex]\( x^2 + 6x + 8 = 0 \)[/tex], follow these detailed steps:
1. Identify the coefficients:
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = 8\)[/tex].
2. Analyze the quadratic equation for factoring:
Factoring a quadratic equation involves writing it as a product of two binomial expressions. Look for two numbers that multiply to [tex]\(ac\)[/tex] (which is [tex]\(1 \cdot 8 = 8\)[/tex]) and add to [tex]\(b\)[/tex] (which is 6).
The numbers that multiply to 8 and add to 6 are 2 and 4.
3. Write the equation in factored form:
Rewrite the middle term, [tex]\(6x\)[/tex], as a sum of two terms using the numbers found in the previous step:
[tex]\[
x^2 + 6x + 8 = x^2 + 2x + 4x + 8
\][/tex]
4. Group and factor by grouping:
Group terms to factor by grouping:
[tex]\[
(x^2 + 2x) + (4x + 8)
\][/tex]
Factor out the greatest common factor from 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 solutions to the equation:
[tex]\[
x + 2 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solving these equations:
[tex]\[
x = -2 \quad \text{or} \quad x = -4
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 + 6x + 8 = 0 \)[/tex] are [tex]\( x = -2 \)[/tex] and [tex]\( x = -4 \)[/tex].
