To solve the quadratic equation [tex]\( x^2 + x - 20 = 0 \)[/tex] by completing the square method, follow these steps:
1. Start with the original equation:
[tex]\[ x^2 + x - 20 = 0 \][/tex]
2. Move the constant term to the right-hand side:
[tex]\[ x^2 + x = 20 \][/tex]
3. To complete the square, we need to add and subtract the same value on the left-hand side of the equation. This value is [tex]\(\left(\frac{b}{2}\right)^2\)[/tex], where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. Here, [tex]\(b = 1\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
4. Add [tex]\(\frac{1}{4}\)[/tex] to both sides:
[tex]\[
x^2 + x + \frac{1}{4} = 20 + \frac{1}{4}
\][/tex]
5. Rewrite the left-hand side as a perfect square and simplify the right-hand side:
[tex]\[
\left(x + \frac{1}{2}\right)^2 = 20 + \frac{1}{4} = \frac{80}{4} + \frac{1}{4} = \frac{81}{4}
\][/tex]
6. Take the square root of both sides:
[tex]\[
x + \frac{1}{2} = \pm \sqrt{\frac{81}{4}}
\][/tex]
7. Simplify the square root expression:
[tex]\[
x + \frac{1}{2} = \pm \frac{9}{2}
\][/tex]
8. Solve for [tex]\(x\)[/tex] by isolating the variable:
[tex]\[
x = -\frac{1}{2} \pm \frac{9}{2}
\][/tex]
9. Calculate the two solutions:
- For the positive case:
[tex]\[
x = -\frac{1}{2} + \frac{9}{2} = \frac{8}{2} = 4
\][/tex]
- For the negative case:
[tex]\[
x = -\frac{1}{2} - \frac{9}{2} = -\frac{10}{2} = -5
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 + x - 20 = 0 \)[/tex] are:
[tex]\[ x = 4 \quad \text{and} \quad x = -5 \][/tex]
