Solve the radical equation 3x−11−−−−−−√+5=x+2.(1 point)

Question

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=−5
x equals negative 5

x=4

Answer 1

To solve the equation \( \sqrt{3x - 11} + 5 = x + 2 \), we can start by isolating the radical term.

1. Subtract 5 from both sides: \[ \sqrt{3x - 11} = x + 2 - 5 \] Simplifying gives: \[ \sqrt{3x - 11} = x - 3 \]

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: \[ 0 = x^2 - 6x + 9 - 3x + 11 \] Combine like terms: \[ 0 = x^2 - 9x + 20 \]

5. This can be rewritten as: \[ x^2 - 9x + 20 = 0 \]

6. Now, factor the quadratic: \[ (x - 4)(x - 5) = 0 \]

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

8. We have potential solutions \( x = 4 \) and \( x = 5 \). We should check for extraneous solutions by substituting back into the original equation.

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

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

Both solutions \( x = 4 \) and \( x = 5 \) are valid.

Thus, the final answer is: x = 4 or x = 5.