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
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.