To solve the equation [tex]\(x^2 - 5x = 6\)[/tex] by factoring, follow these steps:
1. Rewrite the equation:
Start with the given equation:
[tex]\[
x^2 - 5x = 6
\][/tex]
2. Move all terms to one side:
Subtract 6 from both sides to set the equation to zero:
[tex]\[
x^2 - 5x - 6 = 0
\][/tex]
3. Factor the quadratic equation:
Look for two numbers that multiply to give the constant term ([tex]\(-6\)[/tex]) and add to give the coefficient of the linear term ([tex]\(-5\)[/tex]).
These two numbers are [tex]\(-6\)[/tex] and [tex]\(1\)[/tex], because:
[tex]\[
(-6) \cdot 1 = -6 \quad \text{and} \quad (-6) + 1 = -5
\][/tex]
Therefore, we can factor the quadratic as:
[tex]\[
(x - 6)(x + 1) = 0
\][/tex]
4. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 6 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
Solve each equation:
[tex]\[
x = 6 \quad \text{or} \quad x = -1
\][/tex]
So the solutions to the equation [tex]\(x^2 - 5x = 6\)[/tex] are:
[tex]\[
x = \{ -1, 6 \}
\][/tex]
These are the solutions that satisfy the original equation.
