Solve -x^2(x^2-13)=-42

$$- x ^ { 2 } ( x ^ { 2 } - 13 ) = - 42$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{2\left(\sqrt{337}-13\right)}}{2}\approx 1.636697857i$$

$$x=-\frac{i\sqrt{2\left(\sqrt{337}-13\right)}}{2}\approx -0-1.636697857i$$

$$x = -\frac{\sqrt{2 {(\sqrt{337} + 13)}}}{2} \approx -3.959643908$$

$$x = \frac{\sqrt{2 {(\sqrt{337} + 13)}}}{2} \approx 3.959643908$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -x^{2} by x^{2}-13.

$$\left(-x^{2}\right)x^{2}-13\left(-x^{2}\right)=-42$$

Multiply -13 and -1 to get 13.

$$\left(-x^{2}\right)x^{2}+13x^{2}=-42$$

Add 42 to both sides.

$$\left(-x^{2}\right)x^{2}+13x^{2}+42=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-x^{4}+13x^{2}+42=0$$

Substitute t for x^{2}.

$$-t^{2}+13t+42=0$$

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, 13 for b, and 42 for c in the quadratic formula.

$$t=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\times 42}}{-2}$$

Do the calculations.

$$t=\frac{-13±\sqrt{337}}{-2}$$

Solve the equation t=\frac{-13±\sqrt{337}}{-2} when ± is plus and when ± is minus.

$$t=\frac{13-\sqrt{337}}{2} t=\frac{\sqrt{337}+13}{2}$$

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

$$x=-i\sqrt{-\frac{13-\sqrt{337}}{2}} x=i\sqrt{-\frac{13-\sqrt{337}}{2}} x=-\sqrt{\frac{\sqrt{337}+13}{2}} x=\sqrt{\frac{\sqrt{337}+13}{2}}$$

Solve for x

$$x = -\frac{\sqrt{2 {(\sqrt{337} + 13)}}}{2} \approx -3.959643908$$

$$x = \frac{\sqrt{2 {(\sqrt{337} + 13)}}}{2} \approx 3.959643908$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -x^{2} by x^{2}-13.

$$\left(-x^{2}\right)x^{2}-13\left(-x^{2}\right)=-42$$

Multiply -13 and -1 to get 13.

$$\left(-x^{2}\right)x^{2}+13x^{2}=-42$$

Add 42 to both sides.

$$\left(-x^{2}\right)x^{2}+13x^{2}+42=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-x^{4}+13x^{2}+42=0$$

Substitute t for x^{2}.

$$-t^{2}+13t+42=0$$

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, 13 for b, and 42 for c in the quadratic formula.

$$t=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\times 42}}{-2}$$

Do the calculations.

$$t=\frac{-13±\sqrt{337}}{-2}$$

Solve the equation t=\frac{-13±\sqrt{337}}{-2} when ± is plus and when ± is minus.

$$t=\frac{13-\sqrt{337}}{2} t=\frac{\sqrt{337}+13}{2}$$

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

$$x=\frac{\sqrt{2\sqrt{337}+26}}{2} x=-\frac{\sqrt{2\sqrt{337}+26}}{2}$$