Solve y^4-2y^2+5=7

$$y ^ { 4 } - 2 y ^ { 2 } + 5 = 7$$

Solve for y (complex solution)

$$y=i\sqrt{\sqrt{3}-1}\approx 0.855599677i$$

$$y=-i\sqrt{\sqrt{3}-1}\approx -0-0.855599677i$$

$$y=-\sqrt{\sqrt{3}+1}\approx -1.65289165$$

$$y=\sqrt{\sqrt{3}+1}\approx 1.65289165$$

Steps Using the Quadratic Formula

Subtract 7 from both sides.

$$y^{4}-2y^{2}+5-7=0$$

Subtract 7 from 5 to get -2.

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

Substitute t for y^{2}.

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

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 -2 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{2±2\sqrt{3}}{2}$$

Solve the equation t=\frac{2±2\sqrt{3}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{3}+1 t=1-\sqrt{3}$$

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

$$y=-\sqrt{\sqrt{3}+1} y=\sqrt{\sqrt{3}+1} y=-i\sqrt{-\left(1-\sqrt{3}\right)} y=i\sqrt{-\left(1-\sqrt{3}\right)}$$

Solve for y

$$y=-\sqrt{\sqrt{3}+1}\approx -1.65289165$$

$$y=\sqrt{\sqrt{3}+1}\approx 1.65289165$$

Steps Using the Quadratic Formula

Subtract 7 from both sides.

$$y^{4}-2y^{2}+5-7=0$$

Subtract 7 from 5 to get -2.

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

Substitute t for y^{2}.

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

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 -2 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{2±2\sqrt{3}}{2}$$

Solve the equation t=\frac{2±2\sqrt{3}}{2} when ± is plus and when ± is minus.

$$t=\sqrt{3}+1 t=1-\sqrt{3}$$

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

$$y=\sqrt{\sqrt{3}+1} y=-\sqrt{\sqrt{3}+1}$$