Solve -2[-(4:2)-2(3*3^-2):2]^2=x^2+1

$$- 2 [ - ( 4 : 2 ) - 2 ( 3 \times 3 ^ { - 2 } ) : 2 ] ^ { 2 } = x ^ { 2 } + 1$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{107}i}{3}\approx -0-3.448026811i$$

$$x=\frac{\sqrt{107}i}{3}\approx 3.448026811i$$

Steps by Finding Square Root

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

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

Divide 4 by 2 to get 2.

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

Cancel out 2 and 2.

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

Calculate 3 to the power of -1 and get \frac{1}{3}.

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

Subtract \frac{1}{3} from -2 to get -\frac{7}{3}.

$$-2\left(-\frac{7}{3}\right)^{2}=x^{2}+1$$

Calculate -\frac{7}{3} to the power of 2 and get \frac{49}{9}.

$$-2\times \frac{49}{9}=x^{2}+1$$

Multiply -2 and \frac{49}{9} to get -\frac{98}{9}.

$$-\frac{98}{9}=x^{2}+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+1=-\frac{98}{9}$$

Subtract 1 from both sides.

$$x^{2}=-\frac{98}{9}-1$$

Subtract 1 from -\frac{98}{9} to get -\frac{107}{9}.

$$x^{2}=-\frac{107}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{107}i}{3} x=-\frac{\sqrt{107}i}{3}$$

Steps Using the Quadratic Formula

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

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

Divide 4 by 2 to get 2.

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

Cancel out 2 and 2.

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

Calculate 3 to the power of -1 and get \frac{1}{3}.

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

Subtract \frac{1}{3} from -2 to get -\frac{7}{3}.

$$-2\left(-\frac{7}{3}\right)^{2}=x^{2}+1$$

Calculate -\frac{7}{3} to the power of 2 and get \frac{49}{9}.

$$-2\times \frac{49}{9}=x^{2}+1$$

Multiply -2 and \frac{49}{9} to get -\frac{98}{9}.

$$-\frac{98}{9}=x^{2}+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+1=-\frac{98}{9}$$

Add \frac{98}{9} to both sides.

$$x^{2}+1+\frac{98}{9}=0$$

Add 1 and \frac{98}{9} to get \frac{107}{9}.

$$x^{2}+\frac{107}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and \frac{107}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{107}{9}}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{107}{9}}}{2}$$

Multiply -4 times \frac{107}{9}.

$$x=\frac{0±\sqrt{-\frac{428}{9}}}{2}$$

Take the square root of -\frac{428}{9}.

$$x=\frac{0±\frac{2\sqrt{107}i}{3}}{2}$$

Now solve the equation x=\frac{0±\frac{2\sqrt{107}i}{3}}{2} when ± is plus.

$$x=\frac{\sqrt{107}i}{3}$$

Now solve the equation x=\frac{0±\frac{2\sqrt{107}i}{3}}{2} when ± is minus.

$$x=-\frac{\sqrt{107}i}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{107}i}{3} x=-\frac{\sqrt{107}i}{3}$$