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

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

Solve for x_f

$$x_{f}=\frac{\sqrt{6}}{50}\approx 0.048989795$$

$$x_{f}=-\frac{\sqrt{6}}{50}\approx -0.048989795$$

Steps by Finding Square Root

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

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

Divide 8000 by 2 to get 4000.

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

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

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

Calculate 0.02 to the power of 2 and get 0.0004.

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

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

$$2000\left(-x_{f}^{2}\right)+0.8+4=0$$

Add 0.8 and 4 to get 4.8.

$$2000\left(-x_{f}^{2}\right)+4.8=0$$

Subtract 4.8 from both sides. Anything subtracted from zero gives its negation.

$$2000\left(-x_{f}^{2}\right)=-4.8$$

Divide both sides by 2000.

$$-x_{f}^{2}=\frac{-4.8}{2000}$$

Expand \frac{-4.8}{2000} by multiplying both numerator and the denominator by 10.

$$-x_{f}^{2}=\frac{-48}{20000}$$

Reduce the fraction \frac{-48}{20000} to lowest terms by extracting and canceling out 16.

$$-x_{f}^{2}=-\frac{3}{1250}$$

Divide both sides by -1.

$$x_{f}^{2}=\frac{-\frac{3}{1250}}{-1}$$

Express \frac{-\frac{3}{1250}}{-1} as a single fraction.

$$x_{f}^{2}=\frac{-3}{1250\left(-1\right)}$$

Multiply 1250 and -1 to get -1250.

$$x_{f}^{2}=\frac{-3}{-1250}$$

Fraction \frac{-3}{-1250} can be simplified to \frac{3}{1250} by removing the negative sign from both the numerator and the denominator.

$$x_{f}^{2}=\frac{3}{1250}$$

Take the square root of both sides of the equation.

$$x_{f}=\frac{\sqrt{6}}{50} x_{f}=-\frac{\sqrt{6}}{50}$$

Steps Using the Quadratic Formula

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

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

Divide 8000 by 2 to get 4000.

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

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

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

Calculate 0.02 to the power of 2 and get 0.0004.

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

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

$$2000\left(-x_{f}^{2}\right)+0.8+4=0$$

Add 0.8 and 4 to get 4.8.

$$2000\left(-x_{f}^{2}\right)+4.8=0$$

Multiply 2000 and -1 to get -2000.

$$-2000x_{f}^{2}+4.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2000 for a, 0 for b, and 4.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x_{f}=\frac{0±\sqrt{0^{2}-4\left(-2000\right)\times 4.8}}{2\left(-2000\right)}$$

Square 0.

$$x_{f}=\frac{0±\sqrt{-4\left(-2000\right)\times 4.8}}{2\left(-2000\right)}$$

Multiply -4 times -2000.

$$x_{f}=\frac{0±\sqrt{8000\times 4.8}}{2\left(-2000\right)}$$

Multiply 8000 times 4.8.

$$x_{f}=\frac{0±\sqrt{38400}}{2\left(-2000\right)}$$

Take the square root of 38400.

$$x_{f}=\frac{0±80\sqrt{6}}{2\left(-2000\right)}$$

Multiply 2 times -2000.

$$x_{f}=\frac{0±80\sqrt{6}}{-4000}$$

Now solve the equation x_{f}=\frac{0±80\sqrt{6}}{-4000} when ± is plus.

$$x_{f}=-\frac{\sqrt{6}}{50}$$

Now solve the equation x_{f}=\frac{0±80\sqrt{6}}{-4000} when ± is minus.

$$x_{f}=\frac{\sqrt{6}}{50}$$

The equation is now solved.

$$x_{f}=-\frac{\sqrt{6}}{50} x_{f}=\frac{\sqrt{6}}{50}$$