Solve z^4+z^2-12=(z^2+4)^2

$$z ^ { 4 } + z ^ { 2 } - 12 = ( z ^ { 2 } + 4 ) ^ { 2 }$$

Solve for z

$$z=-2i$$

$$z=2i$$

Steps by Finding Square Root

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(z^{2}+4\right)^{2}.

$$z^{4}+z^{2}-12=\left(z^{2}\right)^{2}+8z^{2}+16$$

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

$$z^{4}+z^{2}-12=z^{4}+8z^{2}+16$$

Subtract z^{4} from both sides.

$$z^{4}+z^{2}-12-z^{4}=8z^{2}+16$$

Combine z^{4} and -z^{4} to get 0.

$$z^{2}-12=8z^{2}+16$$

Subtract 8z^{2} from both sides.

$$z^{2}-12-8z^{2}=16$$

Combine z^{2} and -8z^{2} to get -7z^{2}.

$$-7z^{2}-12=16$$

Add 12 to both sides.

$$-7z^{2}=16+12$$

Add 16 and 12 to get 28.

$$-7z^{2}=28$$

Divide both sides by -7.

$$z^{2}=\frac{28}{-7}$$

Divide 28 by -7 to get -4.

$$z^{2}=-4$$

The equation is now solved.

$$z=2i z=-2i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(z^{2}+4\right)^{2}.

$$z^{4}+z^{2}-12=\left(z^{2}\right)^{2}+8z^{2}+16$$

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

$$z^{4}+z^{2}-12=z^{4}+8z^{2}+16$$

Subtract z^{4} from both sides.

$$z^{4}+z^{2}-12-z^{4}=8z^{2}+16$$

Combine z^{4} and -z^{4} to get 0.

$$z^{2}-12=8z^{2}+16$$

Subtract 8z^{2} from both sides.

$$z^{2}-12-8z^{2}=16$$

Combine z^{2} and -8z^{2} to get -7z^{2}.

$$-7z^{2}-12=16$$

Subtract 16 from both sides.

$$-7z^{2}-12-16=0$$

Subtract 16 from -12 to get -28.

$$-7z^{2}-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 0 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{0±\sqrt{0^{2}-4\left(-7\right)\left(-28\right)}}{2\left(-7\right)}$$

Square 0.

$$z=\frac{0±\sqrt{-4\left(-7\right)\left(-28\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$z=\frac{0±\sqrt{28\left(-28\right)}}{2\left(-7\right)}$$

Multiply 28 times -28.

$$z=\frac{0±\sqrt{-784}}{2\left(-7\right)}$$

Take the square root of -784.

$$z=\frac{0±28i}{2\left(-7\right)}$$

Multiply 2 times -7.

$$z=\frac{0±28i}{-14}$$

Now solve the equation z=\frac{0±28i}{-14} when ± is plus.

$$z=-2i$$

Now solve the equation z=\frac{0±28i}{-14} when ± is minus.

$$z=2i$$

The equation is now solved.

$$z=-2i z=2i$$