Solve y^4+2y^2-12=0

$$y ^ { 4 } + 2 y ^ { 2 } - 12 = 0$$

Solve for y (complex solution)

$$y=i\sqrt{\sqrt{13}+1}\approx 2.146054817i$$

$$y=-i\sqrt{\sqrt{13}+1}\approx -0-2.146054817i$$

$$y=-\sqrt{\sqrt{13}-1}\approx -1.614172009$$

$$y=\sqrt{\sqrt{13}-1}\approx 1.614172009$$

Steps Using the Quadratic Formula

Substitute t for y^{2}.

$$t^{2}+2t-12=0$$

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 1 for a, 2 for b, and -12 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 1\left(-12\right)}}{2}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{13}}{2}$$

Solve the equation t=\frac{-2±2\sqrt{13}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{13}-1 t=-\sqrt{13}-1$$

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

$$y=-\sqrt{\sqrt{13}-1} y=\sqrt{\sqrt{13}-1} y=-i\sqrt{\sqrt{13}+1} y=i\sqrt{\sqrt{13}+1}$$

Solve for y

$$y=-\sqrt{\sqrt{13}-1}\approx -1.614172009$$

$$y=\sqrt{\sqrt{13}-1}\approx 1.614172009$$

Steps Using the Quadratic Formula

Substitute t for y^{2}.

$$t^{2}+2t-12=0$$

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 1 for a, 2 for b, and -12 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 1\left(-12\right)}}{2}$$

Do the calculations.

$$t=\frac{-2±2\sqrt{13}}{2}$$

Solve the equation t=\frac{-2±2\sqrt{13}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{13}-1 t=-\sqrt{13}-1$$

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

$$y=\sqrt{\sqrt{13}-1} y=-\sqrt{\sqrt{13}-1}$$