Question

What are the solutions to the equation 2(x2 + x − 11)32 = 54? Check for extraneous solutions. Enter the values in increasing order.

Answer 1

To solve the equation, we can begin by isolating the expression in the parentheses:

2(x^2 + x - 11)^3/2 = 54

Divide both sides by 2:

(x^2 + x - 11)^3/2 = 27

Take the cube root of both sides:

x^2 + x - 11 = 3∛27

Simplify the cube root:

x^2 + x - 11 = 3∛(27) = 3*3 = 9

Rearrange the equation:

x^2 + x - 20 = 0

Factor the quadratic:

(x + 5)(x - 4) = 0

Therefore, the solutions are x = -5 and x = 4.

To check for extraneous solutions, we need to make sure that these values do not make the original equation undefined. Since there are no square roots or denominators, there are no restrictions on x, and both solutions are valid.

So, the solutions to the equation are -5 and 4, in increasing order.