To solve the equation [tex]\( \sqrt{5x + 21} = x + 5 \)[/tex], let's follow these steps:
1. Isolate the Square Root:
The square root is already isolated on the left side:
[tex]\[
\sqrt{5x + 21} = x + 5
\][/tex]
2. Square Both Sides:
To remove the square root, square both sides of the equation:
[tex]\[
\left(\sqrt{5x + 21}\right)^2 = (x + 5)^2
\][/tex]
This simplifies to:
[tex]\[
5x + 21 = x^2 + 10x + 25
\][/tex]
3. Rearrange to Form a Quadratic Equation:
Move all terms to one side to set the equation to zero:
[tex]\[
5x + 21 - 5x = x^2 + 10x + 25 - 5x - 21
\][/tex]
Simplifying this, we get:
[tex]\[
0 = x^2 + 5x + 4
\][/tex]
4. Solve the Quadratic Equation:
Factor the quadratic equation:
[tex]\[
0 = x^2 + 5x + 4 \implies (x + 4)(x + 1) = 0
\][/tex]
Set each factor to zero:
[tex]\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\][/tex]
[tex]\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\][/tex]
5. Verify Each Potential Solution:
It's important to verify that each solution satisfies the original equation, as squaring both sides can introduce extraneous solutions.
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
\sqrt{5(-4) + 21} = -4 + 5
\][/tex]
Simplifies to:
[tex]\[
\sqrt{-20 + 21} = 1 \quad \Rightarrow \quad \sqrt{1} = 1
\][/tex]
Which is true.
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
\sqrt{5(-1) + 21} = -1 + 5
\][/tex]
Simplifies to:
[tex]\[
\sqrt{-5 + 21} = 4 \quad \Rightarrow \quad \sqrt{16} = 4
\][/tex]
Which is true.
Thus, the solutions to the equation [tex]\( \sqrt{5x + 21} = x + 5 \)[/tex] are:
[tex]\[
\boxed{-4 \text{ and } -1}
\][/tex]
