To solve the quadratic equation [tex]\(x^2 - x - 56 = 0\)[/tex], we can use factorization. Here's the step-by-step process:
1. Identify the equation:
[tex]\[
x^2 - x - 56 = 0
\][/tex]
2. Rewrite the quadratic equation in standard form:
The equation is already in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -56\)[/tex].
3. Factor the quadratic equation:
We need to find two numbers that multiply to [tex]\(a \cdot c = 1 \cdot (-56) = -56\)[/tex] and add up to [tex]\(b = -1\)[/tex].
- The two numbers that satisfy these conditions are [tex]\(7\)[/tex] and [tex]\(-8\)[/tex] because [tex]\(7 \times (-8) = -56\)[/tex] and [tex]\(7 + (-8) = -1\)[/tex].
4. Split the middle term using these two numbers:
Rewrite the equation by breaking the middle term [tex]\(-x\)[/tex] into two terms using the numbers found:
[tex]\[
x^2 - 8x + 7x - 56 = 0
\][/tex]
5. Group the terms:
Group the terms in pairs to factor by grouping:
[tex]\[
(x^2 - 8x) + (7x - 56) = 0
\][/tex]
6. Factor out the common factors in each group:
[tex]\[
x(x - 8) + 7(x - 8) = 0
\][/tex]
7. Factor out the common binomial factor:
[tex]\[
(x - 8)(x + 7) = 0
\][/tex]
8. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 8 = 0 \quad \text{or} \quad x + 7 = 0
\][/tex]
Solving these equations, we get:
[tex]\[
x = 8 \quad \text{or} \quad x = -7
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(x^2 - x - 56 = 0\)[/tex] are:
[tex]\[
x = -7 \quad \text{and} \quad x = 8
\][/tex]
