Solve frac{2*(2sqrt{1+3x^2})}{-3x^2}=8/3

$$\frac{ 2 \times (2 \sqrt{ 1+3 { x }^{ 2 } } ) }{ -3 { x }^{ 2 } } = \frac{ 8 }{ 3 }$$

Solve for x (complex solution)

$$x=-\frac{1}{2}i=-0.5i$$

$$x=\frac{1}{2}i=0.5i$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x^{2}, the least common multiple of -3x^{2},3.

$$-2\times 2\sqrt{1+3x^{2}}=8x^{2}$$

Multiply -2 and 2 to get -4.

$$-4\sqrt{1+3x^{2}}=8x^{2}$$

Subtract 8x^{2} from both sides.

$$-4\sqrt{1+3x^{2}}-8x^{2}=0$$

Subtract -8x^{2} from both sides of the equation.

$$-4\sqrt{1+3x^{2}}=8x^{2}$$

Square both sides of the equation.

$$\left(-4\sqrt{1+3x^{2}}\right)^{2}=\left(8x^{2}\right)^{2}$$

Expand \left(-4\sqrt{1+3x^{2}}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{1+3x^{2}}\right)^{2}=\left(8x^{2}\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$16\left(\sqrt{1+3x^{2}}\right)^{2}=\left(8x^{2}\right)^{2}$$

Calculate \sqrt{1+3x^{2}} to the power of 2 and get 1+3x^{2}.

$$16\left(1+3x^{2}\right)=\left(8x^{2}\right)^{2}$$

Use the distributive property to multiply 16 by 1+3x^{2}.

$$16+48x^{2}=\left(8x^{2}\right)^{2}$$

Expand \left(8x^{2}\right)^{2}.

$$16+48x^{2}=8^{2}\left(x^{2}\right)^{2}$$

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

$$16+48x^{2}=8^{2}x^{4}$$

Calculate 8 to the power of 2 and get 64.

$$16+48x^{2}=64x^{4}$$

Subtract 64x^{4} from both sides.

$$16+48x^{2}-64x^{4}=0$$

Substitute t for x^{2}.

$$-64t^{2}+48t+16=0$$

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 -64 for a, 48 for b, and 16 for c in the quadratic formula.

$$t=\frac{-48±\sqrt{48^{2}-4\left(-64\right)\times 16}}{-64\times 2}$$

Do the calculations.

$$t=\frac{-48±80}{-128}$$

Solve the equation t=\frac{-48±80}{-128} when ± is plus and when ± is minus.

$$t=-\frac{1}{4} t=1$$

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

$$x=-\frac{1}{2}i x=\frac{1}{2}i x=-1 x=1$$

Substitute -\frac{1}{2}i for x in the equation \frac{2\times 2\sqrt{1+3x^{2}}}{-3x^{2}}=\frac{8}{3}.

$$\frac{2\times 2\sqrt{1+3\times \left(-\frac{1}{2}i\right)^{2}}}{-3\times \left(-\frac{1}{2}i\right)^{2}}=\frac{8}{3}$$

Simplify. The value x=-\frac{1}{2}i satisfies the equation.

$$\frac{8}{3}=\frac{8}{3}$$

Substitute \frac{1}{2}i for x in the equation \frac{2\times 2\sqrt{1+3x^{2}}}{-3x^{2}}=\frac{8}{3}.

$$\frac{2\times 2\sqrt{1+3\times \left(\frac{1}{2}i\right)^{2}}}{-3\times \left(\frac{1}{2}i\right)^{2}}=\frac{8}{3}$$

Simplify. The value x=\frac{1}{2}i satisfies the equation.

$$\frac{8}{3}=\frac{8}{3}$$

Substitute -1 for x in the equation \frac{2\times 2\sqrt{1+3x^{2}}}{-3x^{2}}=\frac{8}{3}.

$$\frac{2\times 2\sqrt{1+3\left(-1\right)^{2}}}{-3\left(-1\right)^{2}}=\frac{8}{3}$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{8}{3}=\frac{8}{3}$$

Substitute 1 for x in the equation \frac{2\times 2\sqrt{1+3x^{2}}}{-3x^{2}}=\frac{8}{3}.

$$\frac{2\times 2\sqrt{1+3\times 1^{2}}}{-3\times 1^{2}}=\frac{8}{3}$$

Simplify. The value x=1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{8}{3}=\frac{8}{3}$$

List all solutions of -4\sqrt{3x^{2}+1}=8x^{2}.

$$x=-\frac{1}{2}i x=\frac{1}{2}i$$