Solve the radical equation √3x−11+5=x+2

Question

.(1 point)
Responses

x=−5
x equals negative 5

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

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

x=4

Answer 1

To solve the equation \( \sqrt{3x - 11} + 5 = x + 2 \), we can follow these steps:

1. Isolate the square root term: \[ \sqrt{3x - 11} = x + 2 - 5 \] \[ \sqrt{3x - 11} = x - 3 \]

2. 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 to one side: \[ 0 = x^2 - 6x + 9 - 3x + 11 \] \[ 0 = x^2 - 9x + 20 \]

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

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

7. Check each solution in the original equation:

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

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

Thus, the solutions are \( x = 4 \) and \( x = 5 \).

The correct response is: x = 4 or x = 5.