Radical Equations Quick Check

Question

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.