To solve the quadratic equation [tex]\(x^2 + 15x + 54 = 0\)[/tex] by factoring, follow these steps:
1. Identify the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 15\)[/tex], and [tex]\(c = 54\)[/tex].
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] (which is 54) and add up to [tex]\(b\)[/tex] (which is 15):
- We need to find two numbers whose product is [tex]\(54\)[/tex] and whose sum is [tex]\(15\)[/tex].
- The factor pairs of [tex]\(54\)[/tex] are:
- [tex]\(1 \cdot 54\)[/tex]
- [tex]\(2 \cdot 27\)[/tex]
- [tex]\(3 \cdot 18\)[/tex]
- [tex]\(6 \cdot 9\)[/tex]
- [tex]\(9 \cdot 6\)[/tex]
- Among these pairs, [tex]\(9\)[/tex] and [tex]\(6\)[/tex] add up to [tex]\(15\)[/tex]: [tex]\(9 + 6 = 15\)[/tex].
3. Write the middle term [tex]\(15x\)[/tex] as the sum of these two numbers:
[tex]\[
x^2 + 15x + 54 = x^2 + 9x + 6x + 54
\][/tex]
4. Factor by grouping:
- Group the terms in pairs:
[tex]\[
x^2 + 9x + 6x + 54 = (x^2 + 9x) + (6x + 54)
\][/tex]
- Factor out the common factor in each group:
[tex]\[
x(x + 9) + 6(x + 9)
\][/tex]
- Recognize the common binomial factor [tex]\((x + 9)\)[/tex]:
[tex]\[
(x + 9)(x + 6)
\][/tex]
5. Set each factor equal to zero:
[tex]\[
(x + 9)(x + 6) = 0
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- Set each factor to zero and solve:
[tex]\[
x + 9 = 0 \quad \Rightarrow \quad x = -9
\][/tex]
[tex]\[
x + 6 = 0 \quad \Rightarrow \quad x = -6
\][/tex]
So, the solutions to the quadratic equation [tex]\(x^2 + 15x + 54 = 0\)[/tex] are:
[tex]\[
x = -9 \quad \text{and} \quad x = -6
\][/tex]
