To solve the equation [tex]\(\frac{x(x-5)}{6} + 1 = 0\)[/tex], follow these steps:
1. Rewrite the Equation:
Start with the given equation:
[tex]\[
\frac{x(x-5)}{6} + 1 = 0
\][/tex]
2. Eliminate the Fraction:
To eliminate the fraction, multiply every term by 6:
[tex]\[
6 \cdot \left(\frac{x(x-5)}{6}\right) + 6 \cdot 1 = 6 \cdot 0
\][/tex]
This simplifies to:
[tex]\[
x(x-5) + 6 = 0
\][/tex]
3. Simplify and Rearrange:
Distribute [tex]\(x\)[/tex] through the first term and move the constant to the other side of the equation:
[tex]\[
x^2 - 5x + 6 = 0
\][/tex]
4. Factor the Quadratic:
Find two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. Thus, factor the quadratic:
[tex]\[
(x - 2)(x - 3) = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 2 = 0 \quad \text{or} \quad x - 3 = 0
\][/tex]
This gives:
[tex]\[
x = 2 \quad \text{or} \quad x = 3
\][/tex]
Therefore, the solutions to the equation [tex]\(\frac{x(x-5)}{6} + 1 = 0\)[/tex] are:
[tex]\[
x = 2 \quad \text{and} \quad x = 3
\][/tex]
