To solve the equation [tex]\( x^2 + 175 = 0 \)[/tex] by taking square roots, follow the steps below:
1. Isolate [tex]\( x^2 \)[/tex] on one side of the equation:
[tex]\[
x^2 + 175 = 0
\][/tex]
Subtract 175 from both sides:
[tex]\[
x^2 = -175
\][/tex]
2. Take the square root of both sides of the equation:
[tex]\[
x = \pm \sqrt{-175}
\][/tex]
Recall that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit. Therefore:
[tex]\[
x = \pm \sqrt{175} \cdot i
\][/tex]
3. Simplify the expression inside the square root:
First, factor 175 into its prime components:
[tex]\[
175 = 25 \times 7
\][/tex]
Rewrite the square root:
[tex]\[
\sqrt{175} = \sqrt{25 \times 7}
\][/tex]
Use the property of square roots that allows you to take the square root of each factor individually:
[tex]\[
\sqrt{175} = \sqrt{25} \cdot \sqrt{7}
\][/tex]
Since [tex]\(\sqrt{25} = 5\)[/tex]:
[tex]\[
\sqrt{175} = 5\sqrt{7}
\][/tex]
4. Combine this result with the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
x = \pm 5 i \sqrt{7}
\][/tex]
Thus, the complete solutions to the equation [tex]\( x^2 + 175 = 0 \)[/tex] are:
[tex]\[
x = \pm 5 i \sqrt{7}
\][/tex]
Therefore, the correct answer is:
a. [tex]\(\pm 5 i \sqrt{7}\)[/tex]
