Question

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

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

x=−5
x equals negative 5

x=4
x equals 4

x=4 or x=−5

Answer 1

To solve the equation, we can start by isolating the radical term on one side:

3x - 11 - √(x + 5) = x + 2

We can move the x term to the other side:

3x - x - 11 - √(x + 5) = 2

Combine like terms:

2x - 11 - √(x + 5) = 2

Next, we can move the constant term to the other side:

2x - √(x + 5) = 2 + 11

Simplify:

2x - √(x + 5) = 13

Now, we can square both sides of the equation to eliminate the radical:

(2x - √(x + 5))^2 = 13^2

Expand the left side using the distributive property:

4x^2 - 4x√(x + 5) + x + 5 = 169

Combine like terms:

4x^2 - 4x√(x + 5) + x - 164 = 0

Next, we can gather all terms on one side to set up a quadratic equation:

4x^2 - (4x√(x + 5) - x) - 164 = 0

We can simplify the term inside the square root:

4x^2 - (x - 4x√(x + 5)) - 164 = 0

Factor out x from the term inside the square root:

4x^2 - x(1 - 4√(x + 5)) - 164 = 0

Now, we can solve this quadratic equation. Since it is not factorable, we can use the quadratic formula:

x = (-B ± √(B^2 - 4AC)) / (2A)

In this case, A = 4, B = -(1 - 4√(x + 5)), and C = -164. Plugging in these values:

x = (-(1 - 4√(x + 5)) ± √((1 - 4√(x + 5))^2 - 4(4)(-164))) / (2(4))

There is no straightforward way to simplify this expression and solve for x algebraically. Instead, we can use numerical methods or approximations to find the solutions.