Solve by taking square roots.

[tex]\[ x^2 + 175 = 0 \][/tex]

A. [tex]\(\pm 5i \sqrt{7}\)[/tex]
B. [tex]\(\pm 7i \sqrt{5}\)[/tex]
C. [tex]\(\pm \sqrt{175}\)[/tex]
D. [tex]\(\pm 5 \sqrt{5}\)[/tex]



Answer :

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]