Question

Radical Equations Quick Check
4 of 54 of 5 Items
Question
Solve the radical equation 3x−11−−−−−−√+5=x+2
.(1 point)
Responses

x=4
or x=−5
x equals 4 or x equals negative 5

x=4
or x=5
x equals 4 or x equals 5

x=4
x equals 4

x=−5

Answer 1

To solve the equation \( \sqrt{3x - 11} + 5 = x + 2 \):

1. First, isolate the radical:
\[
\sqrt{3x - 11} = x + 2 - 5
\]
\[
\sqrt{3x - 11} = x - 3
\]

2. Next, square both sides to eliminate the square root:
\[
3x - 11 = (x - 3)^2
\]

3. Expand the right side:
\[
3x - 11 = x^2 - 6x + 9
\]

4. Rearrange the equation:
\[
0 = x^2 - 6x + 9 - 3x + 11
\]
\[
0 = x^2 - 9x + 20
\]

5. Factor the quadratic:
\[
0 = (x - 4)(x - 5)
\]

6. Set each factor to zero:
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

Now, we need to check both solutions to see if they satisfy the original equation \( \sqrt{3x - 11} + 5 = x + 2 \):

**For \( x = 4 \)**:
\[
\sqrt{3(4) - 11} + 5 = \sqrt{12 - 11} + 5 = \sqrt{1} + 5 = 1 + 5 = 6
\]
\[
4 + 2 = 6 \quad \Rightarrow \quad \text{True}
\]

**For \( x = 5 \)**:
\[
\sqrt{3(5) - 11} + 5 = \sqrt{15 - 11} + 5 = \sqrt{4} + 5 = 2 + 5 = 7
\]
\[
5 + 2 = 7 \quad \Rightarrow \quad \text{True}
\]

Both solutions check out, therefore the final answers are:
**\( x = 4 \) or \( x = 5 \)**.

So the correct response is:
**x=4 or x=5**.