Solve (x^2+1)^2=60

$$( x ^ { 2 } + 1 ) ^ { 2 } = 60$$

Solve for x (complex solution)

$$x=\sqrt{2\sqrt{15}-1}\approx 2.597299885$$

$$x=-\sqrt{2\sqrt{15}-1}\approx -2.597299885$$

$$x=-i\sqrt{2\sqrt{15}+1}\approx -0-2.95735806i$$

$$x=i\sqrt{2\sqrt{15}+1}\approx 2.95735806i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+1\right)^{2}.

$$\left(x^{2}\right)^{2}+2x^{2}+1=60$$

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

$$x^{4}+2x^{2}+1=60$$

Subtract 60 from both sides.

$$x^{4}+2x^{2}+1-60=0$$

Subtract 60 from 1 to get -59.

$$x^{4}+2x^{2}-59=0$$

Substitute t for x^{2}.

$$t^{2}+2t-59=0$$

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 -59 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 1\left(-59\right)}}{2}$$

Do the calculations.

$$t=\frac{-2±4\sqrt{15}}{2}$$

Solve the equation t=\frac{-2±4\sqrt{15}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{15}-1 t=-2\sqrt{15}-1$$

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

$$x=-\sqrt{2\sqrt{15}-1} x=\sqrt{2\sqrt{15}-1} x=-i\sqrt{2\sqrt{15}+1} x=i\sqrt{2\sqrt{15}+1}$$

Solve for x

$$x=-\sqrt{2\sqrt{15}-1}\approx -2.597299885$$

$$x=\sqrt{2\sqrt{15}-1}\approx 2.597299885$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+1\right)^{2}.

$$\left(x^{2}\right)^{2}+2x^{2}+1=60$$

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

$$x^{4}+2x^{2}+1=60$$

Subtract 60 from both sides.

$$x^{4}+2x^{2}+1-60=0$$

Subtract 60 from 1 to get -59.

$$x^{4}+2x^{2}-59=0$$

Substitute t for x^{2}.

$$t^{2}+2t-59=0$$

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 -59 for c in the quadratic formula.

$$t=\frac{-2±\sqrt{2^{2}-4\times 1\left(-59\right)}}{2}$$

Do the calculations.

$$t=\frac{-2±4\sqrt{15}}{2}$$

Solve the equation t=\frac{-2±4\sqrt{15}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{15}-1 t=-2\sqrt{15}-1$$

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

$$x=\sqrt{2\sqrt{15}-1} x=-\sqrt{2\sqrt{15}-1}$$