To solve the equation \(4 + \sqrt{5x + 66} = x + 10\), follow these steps:
1. Isolate the square root term:
First, isolate the square root on one side of the equation. We can do that by subtracting 4 from both sides:
[tex]\[
\sqrt{5x + 66} = x + 10 - 4
\][/tex]
Simplifying the right side gives us:
[tex]\[
\sqrt{5x + 66} = x + 6
\][/tex]
2. Square both sides of the equation:
To eliminate the square root, square both sides:
[tex]\[
(\sqrt{5x + 66})^2 = (x + 6)^2
\][/tex]
This simplifies to:
[tex]\[
5x + 66 = (x + 6)(x + 6)
\][/tex]
Expanding the right-hand side, we get:
[tex]\[
5x + 66 = x^2 + 12x + 36
\][/tex]
3. Rearrange the equation into a standard quadratic form:
Now, move all terms to one side of the equation to set it equal to zero:
[tex]\[
0 = x^2 + 12x + 36 - 5x - 66
\][/tex]
Simplify this:
[tex]\[
0 = x^2 + 7x - 30
\][/tex]
4. Solve the quadratic equation:
Factor the quadratic equation:
[tex]\[
x^2 + 7x - 30 = 0
\][/tex]
Looking for two numbers that multiply to -30 and add to 7, we find:
[tex]\[
(x + 10)(x - 3) = 0
\][/tex]
Setting each factor equal to zero gives the potential solutions:
[tex]\[
x + 10 = 0 \implies x = -10
\][/tex]
[tex]\[
x - 3 = 0 \implies x = 3
\][/tex]
5. Check the potential solutions in the original equation:
- For \(x = -10\):
[tex]\[
4 + \sqrt{5(-10) + 66} = -10 + 10
\][/tex]
[tex]\[
4 + \sqrt{-50 + 66} = 0
\][/tex]
[tex]\[
4 + \sqrt{16} = 0
\][/tex]
[tex]\[
4 + 4 = 0 \quad \text{(False)}
\][/tex]
So, \(x = -10\) is not a solution.
- For \(x = 3\):
[tex]\[
4 + \sqrt{5(3) + 66} = 3 + 10
\][/tex]
[tex]\[
4 + \sqrt{15 + 66} = 13
\][/tex]
[tex]\[
4 + \sqrt{81} = 13
\][/tex]
[tex]\[
4 + 9 = 13 \quad \text{(True)}
\][/tex]
So, \(x = 3\) is a valid solution.
Hence, the only solution to the equation \(4 + \sqrt{5x + 66} = x + 10\) is:
[tex]\[
\boxed{x = 3}
\][/tex]
