Solve x^4+7=x^2+19

$$x ^ { 4 } + 7 = x ^ { 2 } + 19$$

Solve for x (complex solution)

$$x=2$$

$$x=-2$$

$$x=-\sqrt{3}i\approx -0-1.732050808i$$

$$x=\sqrt{3}i\approx 1.732050808i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x^{4}+7-x^{2}=19$$

Subtract 19 from both sides.

$$x^{4}+7-x^{2}-19=0$$

Subtract 19 from 7 to get -12.

$$x^{4}-12-x^{2}=0$$

Substitute t for x^{2}.

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

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

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

Do the calculations.

$$t=\frac{1±7}{2}$$

Solve the equation t=\frac{1±7}{2} when ± is plus and when ± is minus.

$$t=4 t=-3$$

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

$$x=-2 x=2 x=-\sqrt{3}i x=\sqrt{3}i$$

Solve for x

$$x=-2$$

$$x=2$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x^{4}+7-x^{2}=19$$

Subtract 19 from both sides.

$$x^{4}+7-x^{2}-19=0$$

Subtract 19 from 7 to get -12.

$$x^{4}-12-x^{2}=0$$

Substitute t for x^{2}.

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

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

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

Do the calculations.

$$t=\frac{1±7}{2}$$

Solve the equation t=\frac{1±7}{2} when ± is plus and when ± is minus.

$$t=4 t=-3$$

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

$$x=2 x=-2$$