Question

consider the equation (4x^4 - 38x^3 = 17x^2 + 30x). one of the real solutions to the equation is 10. what are the other solutions, both real and nonreal?

• (0, \frac{-1 + isqrt{11}}{4}, \frac{-1 - isqrt{11}}{4})
• (1, \frac{-1 + isqrt{11}}{4}, \frac{1 - isqrt{11}}{4})
• (1, \frac{1 + isqrt{11}}{4}, \frac{1 - isqrt{11}}{4})
• (0, \frac{1 + isqrt{11}}{4}, \frac{1 - isqrt{11}}{4})

Explanation:

Rearrange the polynomial equation

Using the Roots of Polynomials knowledge point

$$ 4x^4 - 38x^3 - 17x^2 - 30x = 0 $$

Factor out the common term

Using the Factoring Quadratics knowledge point

$$ x(4x^3 - 38x^2 - 17x - 30) = 0 $$

Thus, \(x = 0\) is one of the real solutions.

Perform polynomial division

Using the Polynomial Division knowledge point

$$
LATEXBLOCK0
$$

The quotient is \(4x^2 + 2x + 3 = 0\).

Solve the remaining quadratic equation

Using the Factoring Quadratics knowledge point

$$
LATEXBLOCK1
$$

Identify the complete set of other solutions

The other solutions are:

$$ 0, \frac{-1 + i\sqrt{11}}{4}, \frac{-1 - i\sqrt{11}}{4} $$

Answer:

• (A) \(0, \frac{-1 + i\sqrt{11}}{4}, \frac{-1 - i\sqrt{11}}{4}\) (Correct answer)
• (B) \(1, \frac{-1 + i\sqrt{11}}{4}, \frac{1 - i\sqrt{11}}{4}\)
• (C) \(1, \frac{1 + i\sqrt{11}}{4}, \frac{1 - i\sqrt{11}}{4}\)
• (D) \(0, \frac{1 + i\sqrt{11}}{4}, \frac{1 - i\sqrt{11}}{4}\)