Solve y^4-18y^2+81=(y^2-9)^2

$$y ^ { 4 } - 18 y ^ { 2 } + 81 = ( y ^ { 2 } - 9 ) ^ { 2 }$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y^{2}-9\right)^{2}.

$$y^{4}-18y^{2}+81=\left(y^{2}\right)^{2}-18y^{2}+81$$

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

$$y^{4}-18y^{2}+81=y^{4}-18y^{2}+81$$

Subtract y^{4} from both sides.

$$y^{4}-18y^{2}+81-y^{4}=-18y^{2}+81$$

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

$$-18y^{2}+81=-18y^{2}+81$$

Add 18y^{2} to both sides.

$$-18y^{2}+81+18y^{2}=81$$

Combine -18y^{2} and 18y^{2} to get 0.

$$81=81$$

Compare 81 and 81.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y^{2}-9\right)^{2}.

$$y^{4}-18y^{2}+81=\left(y^{2}\right)^{2}-18y^{2}+81$$

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

$$y^{4}-18y^{2}+81=y^{4}-18y^{2}+81$$

Subtract y^{4} from both sides.

$$y^{4}-18y^{2}+81-y^{4}=-18y^{2}+81$$

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

$$-18y^{2}+81=-18y^{2}+81$$

Add 18y^{2} to both sides.

$$-18y^{2}+81+18y^{2}=81$$

Combine -18y^{2} and 18y^{2} to get 0.

$$81=81$$

Compare 81 and 81.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$