To solve the equation [tex]\(x = \sqrt{x + 30}\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Isolate the square root: The square root is already isolated in the given equation [tex]\(x = \sqrt{x + 30}\)[/tex].
2. Square both sides to eliminate the square root. Squaring both sides of the equation yields:
[tex]\[
x^2 = (\sqrt{x + 30})^2
\][/tex]
Which simplifies to:
[tex]\[
x^2 = x + 30
\][/tex]
3. Rearrange the equation to form a standard quadratic equation. To do this, move all terms to one side of the equation:
[tex]\[
x^2 - x - 30 = 0
\][/tex]
4. Factor the quadratic equation [tex]\(x^2 - x - 30 = 0\)[/tex]. To factor this, we need to find two numbers that multiply to [tex]\(-30\)[/tex] and add up to [tex]\(-1\)[/tex]. Those numbers are [tex]\(6\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[
x^2 - x - 30 = (x - 6)(x + 5) = 0
\][/tex]
5. Set each factor equal to zero to solve for [tex]\(x\)[/tex]:
[tex]\[
x - 6 = 0 \quad \text{or} \quad x + 5 = 0
\][/tex]
6. Solve each equation:
[tex]\[
x - 6 = 0 \implies x = 6
\][/tex]
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
7. Check each potential solution in the original equation to ensure they are valid.
- For [tex]\(x = 6\)[/tex]:
[tex]\[
6 = \sqrt{6 + 30}
\][/tex]
[tex]\[
6 = \sqrt{36}
\][/tex]
[tex]\[
6 = 6
\][/tex]
This is true.
- For [tex]\(x = -5\)[/tex]:
[tex]\[
-5 = \sqrt{-5 + 30}
\][/tex]
[tex]\[
-5 = \sqrt{25}
\][/tex]
[tex]\[
-5 = 5
\][/tex]
This is false, as [tex]\(-5\)[/tex] does not equal [tex]\(5\)[/tex].
Hence, the only valid solution is:
[tex]\[
x = 6
\][/tex]
