Solve x^2(x^2+31)=180

$$x ^ { 2 } ( x ^ { 2 } + 31 ) = 180$$

Solve for x (complex solution)

$$x=\sqrt{5}\approx 2.236067977$$

$$x=-\sqrt{5}\approx -2.236067977$$

$$x=-6i$$

$$x=6i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2} by x^{2}+31.

$$x^{4}+31x^{2}=180$$

Subtract 180 from both sides.

$$x^{4}+31x^{2}-180=0$$

Substitute t for x^{2}.

$$t^{2}+31t-180=0$$

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 1 for a, 31 for b, and -180 for c in the quadratic formula.

$$t=\frac{-31±\sqrt{31^{2}-4\times 1\left(-180\right)}}{2}$$

Do the calculations.

$$t=\frac{-31±41}{2}$$

Solve the equation t=\frac{-31±41}{2} when ± is plus and when ± is minus.

$$t=5 t=-36$$

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

$$x=-\sqrt{5} x=\sqrt{5} x=-6i x=6i$$

Solve for x

$$x=-\sqrt{5}\approx -2.236067977$$

$$x=\sqrt{5}\approx 2.236067977$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2} by x^{2}+31.

$$x^{4}+31x^{2}=180$$

Subtract 180 from both sides.

$$x^{4}+31x^{2}-180=0$$

Substitute t for x^{2}.

$$t^{2}+31t-180=0$$

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 1 for a, 31 for b, and -180 for c in the quadratic formula.

$$t=\frac{-31±\sqrt{31^{2}-4\times 1\left(-180\right)}}{2}$$

Do the calculations.

$$t=\frac{-31±41}{2}$$

Solve the equation t=\frac{-31±41}{2} when ± is plus and when ± is minus.

$$t=5 t=-36$$

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

$$x=\sqrt{5} x=-\sqrt{5}$$