To solve the equation [tex]\( x^2 - 5x = 6 \)[/tex] by factoring, follow these steps:
1. Rewrite the equation in standard quadratic form:
[tex]\[
x^2 - 5x - 6 = 0
\][/tex]
2. Identify and set up the factors:
We need to factorize the quadratic expression [tex]\( x^2 - 5x - 6 \)[/tex]. This expression can be factored into the form [tex]\((x - a)(x - b) = 0\)[/tex].
3. Find two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add up to [tex]\(-5\)[/tex] (the coefficient of [tex]\(x\)[/tex]):
These two numbers are [tex]\(-6\)[/tex] and [tex]\(1\)[/tex] because:
[tex]\[
-6 \times 1 = -6
\][/tex]
[tex]\[
-6 + 1 = -5
\][/tex]
4. Write the factored form:
Using the numbers found, we can write:
[tex]\[
(x - 6)(x + 1) = 0
\][/tex]
5. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
For the first factor:
[tex]\[
x - 6 = 0 \implies x = 6
\][/tex]
For the second factor:
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
6. List the solutions:
The solutions to the equation [tex]\( x^2 - 5x - 6 = 0 \)[/tex] are [tex]\( x = 6 \)[/tex] and [tex]\( x = -1 \)[/tex].
Thus, the solutions are:
[tex]\[
x = 6
\][/tex]
[tex]\[
x = -1
\][/tex]
