To solve the quadratic equation [tex]\(x^2 + 12x - 11 = 0\)[/tex] by completing the square, follow these steps:
1. Start with the given equation:
[tex]\[
x^2 + 12x - 11 = 0
\][/tex]
2. Move the constant term to the right side of the equation:
[tex]\[
x^2 + 12x = 11
\][/tex]
3. Find the value that completes the square:
We need to add and subtract [tex]\(\left(\frac{b}{2}\right)^2\)[/tex] inside the equation, where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]. In this case, [tex]\(b = 12\)[/tex].
[tex]\[
\left(\frac{12}{2}\right)^2 = 6^2 = 36
\][/tex]
4. Add and subtract this value on the left side of the equation:
[tex]\[
x^2 + 12x + 36 - 36 = 11
\][/tex]
[tex]\[
x^2 + 12x + 36 = 47
\][/tex]
5. Rewrite the left side as a perfect square trinomial:
[tex]\[
(x + 6)^2 = 47
\][/tex]
6. Take the square root of both sides:
[tex]\[
x + 6 = \pm\sqrt{47}
\][/tex]
7. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -6 \pm \sqrt{47}
\][/tex]
8. Calculate the numerical values and round to three significant figures:
- The positive solution:
[tex]\[
x_1 = -6 + \sqrt{47} \approx 0.856
\][/tex]
- The negative solution:
[tex]\[
x_2 = -6 - \sqrt{47} \approx -12.856
\][/tex]
Therefore, the solutions to the equation [tex]\(x^2 + 12x - 11 = 0\)[/tex] are:
[tex]\[
x \approx 0.856 \quad \text{and} \quad x \approx -12.856
\][/tex]
rounded to three significant figures.
