To solve the equation [tex]\(-2(x + 9)^2 + 2 = -12\)[/tex] using square roots, follow these steps:
1. Rearrange the equation: First, we need to isolate the squared term.
[tex]\[
-2(x + 9)^2 + 2 = -12
\][/tex]
Subtract 2 from both sides to start isolating the squared term:
[tex]\[
-2(x + 9)^2 = -12 - 2
\][/tex]
This simplifies to:
[tex]\[
-2(x + 9)^2 = -14
\][/tex]
2. Isolate [tex]\((x + 9)^2\)[/tex]: Divide both sides by -2 to solve for [tex]\((x + 9)^2\)[/tex]:
[tex]\[
(x + 9)^2 = \frac{-14}{-2}
\][/tex]
Which simplifies to:
[tex]\[
(x + 9)^2 = 7
\][/tex]
3. Take the square root of both sides: To eliminate the square, we take the square root of both sides of the equation:
[tex]\[
x + 9 = \pm\sqrt{7}
\][/tex]
This yields two equations:
[tex]\[
x + 9 = \sqrt{7}
\][/tex]
and
[tex]\[
x + 9 = -\sqrt{7}
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Subtract 9 from both sides in each case to find the solutions for [tex]\(x\)[/tex].
For the first equation, we get:
[tex]\[
x = \sqrt{7} - 9
\][/tex]
For the second equation, we get:
[tex]\[
x = -\sqrt{7} - 9
\][/tex]
5. Calculate the numerical values and round them: Finally, we calculate the numerical values for [tex]\(\sqrt{7}\)[/tex] and round the results to one decimal place.
[tex]\[
x_1 = \sqrt{7} - 9 \approx -6.4
\][/tex]
[tex]\[
x_2 = -\sqrt{7} - 9 \approx -11.6
\][/tex]
Thus, the solutions to the equation [tex]\(-2(x + 9)^2 + 2 = -12\)[/tex] rounded to one decimal place are:
[tex]\[ x = \{ -6.4, -11.6 \} \][/tex]
