Solve -1/2*80/0.02(0.02^2-x_f^2)+4=0

$$- \frac { 1 } { 2 } \cdot \frac { 80 } { 0.02 } ( 0.02 ^ { 2 } - x _ { f } ^ { 2 } ) + 4 = 0$$

Solve for x_f

$$x_{f}=-0.04i$$

$$x_{f}=0.04i$$

Steps by Finding Square Root

Expand \frac{80}{0.02} by multiplying both numerator and the denominator by 100.

$$-\frac{1}{2}\times \frac{8000}{2}\left(0.02^{2}-x_{f}^{2}\right)+4=0$$

Divide 8000 by 2 to get 4000.

$$-\frac{1}{2}\times 4000\left(0.02^{2}-x_{f}^{2}\right)+4=0$$

Multiply -\frac{1}{2} and 4000 to get -2000.

$$-2000\left(0.02^{2}-x_{f}^{2}\right)+4=0$$

Calculate 0.02 to the power of 2 and get 0.0004.

$$-2000\left(0.0004-x_{f}^{2}\right)+4=0$$

Use the distributive property to multiply -2000 by 0.0004-x_{f}^{2}.

$$-0.8+2000x_{f}^{2}+4=0$$

Add -0.8 and 4 to get 3.2.

$$3.2+2000x_{f}^{2}=0$$

Subtract 3.2 from both sides. Anything subtracted from zero gives its negation.

$$2000x_{f}^{2}=-3.2$$

Divide both sides by 2000.

$$x_{f}^{2}=\frac{-3.2}{2000}$$

Expand \frac{-3.2}{2000} by multiplying both numerator and the denominator by 10.

$$x_{f}^{2}=\frac{-32}{20000}$$

Reduce the fraction \frac{-32}{20000} to lowest terms by extracting and canceling out 32.

$$x_{f}^{2}=-\frac{1}{625}$$

The equation is now solved.

$$x_{f}=\frac{1}{25}i x_{f}=-\frac{1}{25}i$$

Steps Using the Quadratic Formula

Expand \frac{80}{0.02} by multiplying both numerator and the denominator by 100.

$$-\frac{1}{2}\times \frac{8000}{2}\left(0.02^{2}-x_{f}^{2}\right)+4=0$$

Divide 8000 by 2 to get 4000.

$$-\frac{1}{2}\times 4000\left(0.02^{2}-x_{f}^{2}\right)+4=0$$

Multiply -\frac{1}{2} and 4000 to get -2000.

$$-2000\left(0.02^{2}-x_{f}^{2}\right)+4=0$$

Calculate 0.02 to the power of 2 and get 0.0004.

$$-2000\left(0.0004-x_{f}^{2}\right)+4=0$$

Use the distributive property to multiply -2000 by 0.0004-x_{f}^{2}.

$$-0.8+2000x_{f}^{2}+4=0$$

Add -0.8 and 4 to get 3.2.

$$3.2+2000x_{f}^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$2000x_{f}^{2}+3.2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2000 for a, 0 for b, and 3.2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x_{f}=\frac{0±\sqrt{0^{2}-4\times 2000\times 3.2}}{2\times 2000}$$

Square 0.

$$x_{f}=\frac{0±\sqrt{-4\times 2000\times 3.2}}{2\times 2000}$$

Multiply -4 times 2000.

$$x_{f}=\frac{0±\sqrt{-8000\times 3.2}}{2\times 2000}$$

Multiply -8000 times 3.2.

$$x_{f}=\frac{0±\sqrt{-25600}}{2\times 2000}$$

Take the square root of -25600.

$$x_{f}=\frac{0±160i}{2\times 2000}$$

Multiply 2 times 2000.

$$x_{f}=\frac{0±160i}{4000}$$

Now solve the equation x_{f}=\frac{0±160i}{4000} when ± is plus.

$$x_{f}=\frac{1}{25}i$$

Now solve the equation x_{f}=\frac{0±160i}{4000} when ± is minus.

$$x_{f}=-\frac{1}{25}i$$

The equation is now solved.

$$x_{f}=\frac{1}{25}i x_{f}=-\frac{1}{25}i$$