Solve x^2(3x^2+7)=4

$$x ^ { 2 } ( 3 x ^ { 2 } + 7 ) = 4$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{6\left(\sqrt{97}+7\right)}}{6}\approx 1.675751463i$$

$$x=-\frac{i\sqrt{6\left(\sqrt{97}+7\right)}}{6}\approx -0-1.675751463i$$

$$x=-\frac{\sqrt{6\left(\sqrt{97}-7\right)}}{6}\approx -0.689064317$$

$$x=\frac{\sqrt{6\left(\sqrt{97}-7\right)}}{6}\approx 0.689064317$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2} by 3x^{2}+7.

$$3x^{4}+7x^{2}=4$$

Subtract 4 from both sides.

$$3x^{4}+7x^{2}-4=0$$

Substitute t for x^{2}.

$$3t^{2}+7t-4=0$$

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 3 for a, 7 for b, and -4 for c in the quadratic formula.

$$t=\frac{-7±\sqrt{7^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-7±\sqrt{97}}{6}$$

Solve the equation t=\frac{-7±\sqrt{97}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{97}-7}{6} t=\frac{-\sqrt{97}-7}{6}$$

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

$$x=-\sqrt{\frac{\sqrt{97}-7}{6}} x=\sqrt{\frac{\sqrt{97}-7}{6}} x=-i\sqrt{\frac{\sqrt{97}+7}{6}} x=i\sqrt{\frac{\sqrt{97}+7}{6}}$$

Solve for x

$$x=-\frac{\sqrt{6\left(\sqrt{97}-7\right)}}{6}\approx -0.689064317$$

$$x=\frac{\sqrt{6\left(\sqrt{97}-7\right)}}{6}\approx 0.689064317$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2} by 3x^{2}+7.

$$3x^{4}+7x^{2}=4$$

Subtract 4 from both sides.

$$3x^{4}+7x^{2}-4=0$$

Substitute t for x^{2}.

$$3t^{2}+7t-4=0$$

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 3 for a, 7 for b, and -4 for c in the quadratic formula.

$$t=\frac{-7±\sqrt{7^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Do the calculations.

$$t=\frac{-7±\sqrt{97}}{6}$$

Solve the equation t=\frac{-7±\sqrt{97}}{6} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{97}-7}{6} t=\frac{-\sqrt{97}-7}{6}$$

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

$$x=\frac{\sqrt{\frac{2\sqrt{97}-14}{3}}}{2} x=-\frac{\sqrt{\frac{2\sqrt{97}-14}{3}}}{2}$$