Find all valid real roots for the equation.
-$$3 x ^ { 8 } + 4 x ^ { 4 } = 32$$
A) no real roots
B) $x = \sqrt [ 4 ] { 4 }$
C) $x = \pm \frac { \sqrt [ 4 ] { 216 } } { 3 } , x = \sqrt [ 4 ] { 4 }$
D) $x = \pm \frac { \sqrt [ 4 ] { 216 } } { 3 }$

Question

Quizplus · Accepted Answer

The equation can be rewritten as $3(x^4)^2 + 4x^4 - 32 = 0$. Let $y = x^4$, then $3y^2 + 4y - 32 = 0$. Solving this quadratic equation gives $y = \frac{8}{3}$ or $y = -4$. Since $y = x^4$ must be non-negative, only $y = \frac{8}{3}$ is valid. Thus, $x^4 = \frac{8}{3}$, giving $x = \pm \frac{\sqrt[4]{216}}{3}$.

Find All Valid Real Roots for the Equation 3x8+4x4=323 x ^ { 8 } + 4 x ^ { 4 } = 323x8+4x4=32

Find All Valid Real Roots for the Equation $3 x ^ { 8 } + 4 x ^ { 4 } = 32$