Solve 3x^4+64=52x^2

$$3 { x }^{ 4 } +64=52 { x }^{ 2 }$$

Solve for x

$$x=-4$$

$$x=4$$

$$x = \frac{2 \sqrt{3}}{3} \approx 1.154700538$$

$$x = -\frac{2 \sqrt{3}}{3} \approx -1.154700538$$

Steps Using the Quadratic Formula

Subtract 52x^{2} from both sides.

$$3x^{4}+64-52x^{2}=0$$

Substitute t for x^{2}.

$$3t^{2}-52t+64=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 3 for a, -52 for b, and 64 for c in the quadratic formula.

$$t=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 3\times 64}}{2\times 3}$$

Do the calculations.

$$t=\frac{52±44}{6}$$

Solve the equation t=\frac{52±44}{6} when ± is plus and when ± is minus.

$$t=16 t=\frac{4}{3}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=4 x=-4 x=\frac{2\sqrt{3}}{3} x=-\frac{2\sqrt{3}}{3}$$