Solve (x^2+6)(7-x^2)-36=x^4+12x^2

$$( x ^ { 2 } + 6 ) ( 7 - x ^ { 2 } ) - 36 = x ^ { 4 } + 12 x ^ { 2 }$$

Solve for x (complex solution)

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

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

$$x=-\frac{\sqrt{2}}{2}\approx -0.707106781$$

$$x=\frac{\sqrt{2}}{2}\approx 0.707106781$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2}+6 by 7-x^{2} and combine like terms.

$$x^{2}-x^{4}+42-36=x^{4}+12x^{2}$$

Subtract 36 from 42 to get 6.

$$x^{2}-x^{4}+6=x^{4}+12x^{2}$$

Subtract x^{4} from both sides.

$$x^{2}-x^{4}+6-x^{4}=12x^{2}$$

Combine -x^{4} and -x^{4} to get -2x^{4}.

$$x^{2}-2x^{4}+6=12x^{2}$$

Subtract 12x^{2} from both sides.

$$x^{2}-2x^{4}+6-12x^{2}=0$$

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}-2x^{4}+6=0$$

Substitute t for x^{2}.

$$-2t^{2}-11t+6=0$$

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 -2 for a, -11 for b, and 6 for c in the quadratic formula.

$$t=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-2\right)\times 6}}{-2\times 2}$$

Do the calculations.

$$t=\frac{11±13}{-4}$$

Solve the equation t=\frac{11±13}{-4} when ± is plus and when ± is minus.

$$t=-6 t=\frac{1}{2}$$

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

$$x=-\sqrt{6}i x=\sqrt{6}i x=-\frac{\sqrt{2}}{2} x=\frac{\sqrt{2}}{2}$$

Solve for x

$$x=-\frac{\sqrt{2}}{2}\approx -0.707106781$$

$$x=\frac{\sqrt{2}}{2}\approx 0.707106781$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2}+6 by 7-x^{2} and combine like terms.

$$x^{2}-x^{4}+42-36=x^{4}+12x^{2}$$

Subtract 36 from 42 to get 6.

$$x^{2}-x^{4}+6=x^{4}+12x^{2}$$

Subtract x^{4} from both sides.

$$x^{2}-x^{4}+6-x^{4}=12x^{2}$$

Combine -x^{4} and -x^{4} to get -2x^{4}.

$$x^{2}-2x^{4}+6=12x^{2}$$

Subtract 12x^{2} from both sides.

$$x^{2}-2x^{4}+6-12x^{2}=0$$

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}-2x^{4}+6=0$$

Substitute t for x^{2}.

$$-2t^{2}-11t+6=0$$

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 -2 for a, -11 for b, and 6 for c in the quadratic formula.

$$t=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-2\right)\times 6}}{-2\times 2}$$

Do the calculations.

$$t=\frac{11±13}{-4}$$

Solve the equation t=\frac{11±13}{-4} when ± is plus and when ± is minus.

$$t=-6 t=\frac{1}{2}$$

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

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