To solve the equation [tex]\(5x^2 + 1 = 51\)[/tex] by finding square roots, follow these steps:
1. Isolate the quadratic term:
[tex]\[
5x^2 + 1 = 51
\][/tex]
Subtract 1 from both sides to isolate the term with [tex]\(x^2\)[/tex]:
[tex]\[
5x^2 + 1 - 1 = 51 - 1 \implies 5x^2 = 50
\][/tex]
2. Solve for [tex]\(x^2\)[/tex]:
Divide both sides of the equation by 5 to solve for [tex]\(x^2\)[/tex]:
[tex]\[
x^2 = \frac{50}{5} \implies x^2 = 10
\][/tex]
3. Take the square root of both sides:
Find the square roots to solve for [tex]\(x\)[/tex]. There are two possible solutions, the positive and the negative square roots:
[tex]\[
x = \sqrt{10} \quad \text{and} \quad x = -\sqrt{10}
\][/tex]
So, the solutions to the equation [tex]\(5x^2 + 1 = 51\)[/tex] are:
[tex]\[
x = \sqrt{10} \approx 3.1622776601683795 \quad \text{and} \quad x = -\sqrt{10} \approx -3.1622776601683795
\][/tex]
Thus, the correct option from the given choices is [tex]\(x = \sqrt{10}, -\sqrt{10}\)[/tex].
