Solve 12x^{2}-12x^4=25-22^2x^4-220x^2

$$12 x ^ { 2 } - 12 x ^ { 4 } = 25 - 22 ^ { 2 } x ^ { 4 } - 220 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{59\left(\sqrt{6314}+58\right)}}{118}\approx 0.76319101i$$

$$x=-\frac{i\sqrt{59\left(\sqrt{6314}+58\right)}}{118}\approx -0-0.76319101i$$

$$x=-\frac{\sqrt{59\left(\sqrt{6314}-58\right)}}{118}\approx -0.301554462$$

$$x=\frac{\sqrt{59\left(\sqrt{6314}-58\right)}}{118}\approx 0.301554462$$

Steps Using the Quadratic Formula

Calculate 22 to the power of 2 and get 484.

$$12x^{2}-12x^{4}=25-484x^{4}-220x^{2}$$

Subtract 25 from both sides.

$$12x^{2}-12x^{4}-25=-484x^{4}-220x^{2}$$

Add 484x^{4} to both sides.

$$12x^{2}-12x^{4}-25+484x^{4}=-220x^{2}$$

Combine -12x^{4} and 484x^{4} to get 472x^{4}.

$$12x^{2}+472x^{4}-25=-220x^{2}$$

Add 220x^{2} to both sides.

$$12x^{2}+472x^{4}-25+220x^{2}=0$$

Combine 12x^{2} and 220x^{2} to get 232x^{2}.

$$232x^{2}+472x^{4}-25=0$$

Substitute t for x^{2}.

$$472t^{2}+232t-25=0$$

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 472 for a, 232 for b, and -25 for c in the quadratic formula.

$$t=\frac{-232±\sqrt{232^{2}-4\times 472\left(-25\right)}}{2\times 472}$$

Do the calculations.

$$t=\frac{-232±4\sqrt{6314}}{944}$$

Solve the equation t=\frac{-232±4\sqrt{6314}}{944} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{6314}}{236}-\frac{29}{118} t=-\frac{\sqrt{6314}}{236}-\frac{29}{118}$$

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

$$x=-\sqrt{\frac{\sqrt{6314}}{236}-\frac{29}{118}} x=\sqrt{\frac{\sqrt{6314}}{236}-\frac{29}{118}} x=-i\sqrt{\frac{\sqrt{6314}}{236}+\frac{29}{118}} x=i\sqrt{\frac{\sqrt{6314}}{236}+\frac{29}{118}}$$

Solve for x

$$x=-\frac{\sqrt{59\left(\sqrt{6314}-58\right)}}{118}\approx -0.301554462$$

$$x=\frac{\sqrt{59\left(\sqrt{6314}-58\right)}}{118}\approx 0.301554462$$

Steps Using the Quadratic Formula

Calculate 22 to the power of 2 and get 484.

$$12x^{2}-12x^{4}=25-484x^{4}-220x^{2}$$

Subtract 25 from both sides.

$$12x^{2}-12x^{4}-25=-484x^{4}-220x^{2}$$

Add 484x^{4} to both sides.

$$12x^{2}-12x^{4}-25+484x^{4}=-220x^{2}$$

Combine -12x^{4} and 484x^{4} to get 472x^{4}.

$$12x^{2}+472x^{4}-25=-220x^{2}$$

Add 220x^{2} to both sides.

$$12x^{2}+472x^{4}-25+220x^{2}=0$$

Combine 12x^{2} and 220x^{2} to get 232x^{2}.

$$232x^{2}+472x^{4}-25=0$$

Substitute t for x^{2}.

$$472t^{2}+232t-25=0$$

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 472 for a, 232 for b, and -25 for c in the quadratic formula.

$$t=\frac{-232±\sqrt{232^{2}-4\times 472\left(-25\right)}}{2\times 472}$$

Do the calculations.

$$t=\frac{-232±4\sqrt{6314}}{944}$$

Solve the equation t=\frac{-232±4\sqrt{6314}}{944} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{6314}}{236}-\frac{29}{118} t=-\frac{\sqrt{6314}}{236}-\frac{29}{118}$$

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

$$x=\frac{\sqrt{\frac{\sqrt{6314}-58}{59}}}{2} x=-\frac{\sqrt{\frac{\sqrt{6314}-58}{59}}}{2}$$