To solve the equation [tex]\( x = \sqrt{2x + 15} \)[/tex], we will go through the following steps:
1. Square both sides of the equation to eliminate the square root.
[tex]\[
(\sqrt{2x + 15})^2 = x^2
\][/tex]
This simplifies to:
[tex]\[
2x + 15 = x^2
\][/tex]
2. Rearrange the equation to form a standard quadratic equation. To do this, move all terms to one side of the equation:
[tex]\[
x^2 - 2x - 15 = 0
\][/tex]
3. Factorize the quadratic equation. We look for two numbers that multiply to [tex]\(-15\)[/tex] (the constant term) and add up to [tex]\(-2\)[/tex] (the coefficient of [tex]\(x\)[/tex]). These numbers are [tex]\(-5\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
x^2 - 2x - 15 = (x - 5)(x + 3) = 0
\][/tex]
4. Set each factor to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 5 = 0 \quad \text{or} \quad x + 3 = 0
\][/tex]
[tex]\[
x = 5 \quad \text{or} \quad x = -3
\][/tex]
5. Verify each potential solution by substituting it back into the original equation [tex]\( x = \sqrt{2x + 15} \)[/tex].
For [tex]\( x = 5 \)[/tex]:
[tex]\[
5 = \sqrt{2(5) + 15}
\][/tex]
[tex]\[
5 = \sqrt{10 + 15}
\][/tex]
[tex]\[
5 = \sqrt{25}
\][/tex]
[tex]\[
5 = 5
\][/tex]
This is true.
For [tex]\( x = -3 \)[/tex]:
[tex]\[
-3 = \sqrt{2(-3) + 15}
\][/tex]
[tex]\[
-3 = \sqrt{-6 + 15}
\][/tex]
[tex]\[
-3 = \sqrt{9}
\][/tex]
[tex]\[
-3 = 3
\][/tex]
This is false.
6. Conclusion: The only valid solution is [tex]\( x = 5 \)[/tex]. The value [tex]\( x = -3 \)[/tex] does not satisfy the original equation since we can't take the square root of a number and get a negative result (as square roots are non-negative by definition).
Thus, the solution to the equation [tex]\( x = \sqrt{2x + 15} \)[/tex] is:
[tex]\[
x = 5
\][/tex]
