Solve left(2x^2+21right)^2=left(x^2+22right)left(x^2-22right)

$${ \left(2 { x }^{ 2 } +21 \right) }^{ 2 } = \left( { x }^{ 2 } +22 \right) \left( { x }^{ 2 } -22 \right)$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{24975}e^{\frac{\left(\arctan(\frac{\sqrt{1011}}{42})+\pi \right)i}{2}}}{3}\approx -1.334058267+3.972368495i$$

$$x=\frac{\sqrt[4]{24975}e^{\frac{\arctan(\frac{\sqrt{1011}}{42})i+3\pi i}{2}}}{3}\approx 1.334058267-3.972368495i$$

$$x=\frac{\sqrt[4]{24975}e^{\frac{-\arctan(\frac{\sqrt{1011}}{42})i+3\pi i}{2}}}{3}\approx -1.334058267-3.972368495i$$

$$x=\frac{\sqrt[4]{24975}e^{\frac{-\arctan(\frac{\sqrt{1011}}{42})i+\pi i}{2}}}{3}\approx 1.334058267+3.972368495i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x^{2}+21\right)^{2}.

$$4\left(x^{2}\right)^{2}+84x^{2}+441=\left(x^{2}+22\right)\left(x^{2}-22\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4x^{4}+84x^{2}+441=\left(x^{2}+22\right)\left(x^{2}-22\right)$$

Consider \left(x^{2}+22\right)\left(x^{2}-22\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 22.

$$4x^{4}+84x^{2}+441=\left(x^{2}\right)^{2}-484$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4x^{4}+84x^{2}+441=x^{4}-484$$

Subtract x^{4} from both sides.

$$4x^{4}+84x^{2}+441-x^{4}=-484$$

Combine 4x^{4} and -x^{4} to get 3x^{4}.

$$3x^{4}+84x^{2}+441=-484$$

Add 484 to both sides.

$$3x^{4}+84x^{2}+441+484=0$$

Add 441 and 484 to get 925.

$$3x^{4}+84x^{2}+925=0$$

Substitute t for x^{2}.

$$3t^{2}+84t+925=0$$

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, 84 for b, and 925 for c in the quadratic formula.

$$t=\frac{-84±\sqrt{84^{2}-4\times 3\times 925}}{2\times 3}$$

Do the calculations.

$$t=\frac{-84±\sqrt{-4044}}{6}$$

Solve the equation t=\frac{-84±\sqrt{-4044}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{1011}i}{3}-14 t=-\frac{\sqrt{1011}i}{3}-14$$

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

$$x=\frac{\sqrt[4]{24975}e^{\frac{-\arctan(\frac{\sqrt{1011}}{42})i+3\pi i}{2}}}{3} x=\frac{\sqrt[4]{24975}e^{\frac{-\arctan(\frac{\sqrt{1011}}{42})i+\pi i}{2}}}{3} x=\frac{\sqrt[4]{24975}e^{\frac{\arctan(\frac{\sqrt{1011}}{42})i+3\pi i}{2}}}{3} x=\frac{\sqrt[4]{24975}e^{\frac{\left(\arctan(\frac{\sqrt{1011}}{42})+\pi \right)i}{2}}}{3}$$