Solve (x^2+2)(x^2-12)=72

$$( { x }^{ 2 } +2)( { x }^{ 2 } -12)=72$$

Solve for x (complex solution)

$$x=4$$

$$x=-4$$

$$x=-\sqrt{6}i\approx -0-2.449489743i$$

$$x=\sqrt{6}i\approx 2.449489743i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2}+2 by x^{2}-12 and combine like terms.

$$x^{4}-10x^{2}-24=72$$

Subtract 72 from both sides.

$$x^{4}-10x^{2}-24-72=0$$

Subtract 72 from -24 to get -96.

$$x^{4}-10x^{2}-96=0$$

Substitute t for x^{2}.

$$t^{2}-10t-96=0$$

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, -10 for b, and -96 for c in the quadratic formula.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\left(-96\right)}}{2}$$

Do the calculations.

$$t=\frac{10±22}{2}$$

Solve the equation t=\frac{10±22}{2} when ± is plus and when ± is minus.

$$t=16 t=-6$$

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

$$x=-4 x=4 x=-\sqrt{6}i x=\sqrt{6}i$$

Solve for x

$$x=-4$$

$$x=4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2}+2 by x^{2}-12 and combine like terms.

$$x^{4}-10x^{2}-24=72$$

Subtract 72 from both sides.

$$x^{4}-10x^{2}-24-72=0$$

Subtract 72 from -24 to get -96.

$$x^{4}-10x^{2}-96=0$$

Substitute t for x^{2}.

$$t^{2}-10t-96=0$$

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, -10 for b, and -96 for c in the quadratic formula.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\left(-96\right)}}{2}$$

Do the calculations.

$$t=\frac{10±22}{2}$$

Solve the equation t=\frac{10±22}{2} when ± is plus and when ± is minus.

$$t=16 t=-6$$

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

$$x=4 x=-4$$