Solve 2/80x^2+4/25(1-x)^2-0.2x=0

$$\frac { 2 } { 80 } x ^ { 2 } + \frac { 4 } { 25 } ( 1 - x ) ^ { 2 } - 0.2 x = 0$$

Solve for x

$$x = \frac{4 \sqrt{95} + 52}{37} \approx 2.459112902$$

$$x=\frac{52-4\sqrt{95}}{37}\approx 0.351697909$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{2}{80} to lowest terms by extracting and canceling out 2.

$$\frac{1}{40}x^{2}+\frac{4}{25}\left(1-x\right)^{2}-0.2x=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

$$\frac{1}{40}x^{2}+\frac{4}{25}\left(1-2x+x^{2}\right)-0.2x=0$$

Use the distributive property to multiply \frac{4}{25} by 1-2x+x^{2}.

$$\frac{1}{40}x^{2}+\frac{4}{25}-\frac{8}{25}x+\frac{4}{25}x^{2}-0.2x=0$$

Combine \frac{1}{40}x^{2} and \frac{4}{25}x^{2} to get \frac{37}{200}x^{2}.

$$\frac{37}{200}x^{2}+\frac{4}{25}-\frac{8}{25}x-0.2x=0$$

Combine -\frac{8}{25}x and -0.2x to get -\frac{13}{25}x.

$$\frac{37}{200}x^{2}+\frac{4}{25}-\frac{13}{25}x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{37}{200}x^{2}-\frac{13}{25}x+\frac{4}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{37}{200} for a, -\frac{13}{25} for b, and \frac{4}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\left(-\frac{13}{25}\right)^{2}-4\times \frac{37}{200}\times \frac{4}{25}}}{2\times \frac{37}{200}}$$

Square -\frac{13}{25} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{169}{625}-4\times \frac{37}{200}\times \frac{4}{25}}}{2\times \frac{37}{200}}$$

Multiply -4 times \frac{37}{200}.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{169}{625}-\frac{37}{50}\times \frac{4}{25}}}{2\times \frac{37}{200}}$$

Multiply -\frac{37}{50} times \frac{4}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{169-74}{625}}}{2\times \frac{37}{200}}$$

Add \frac{169}{625} to -\frac{74}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{19}{125}}}{2\times \frac{37}{200}}$$

Take the square root of \frac{19}{125}.

$$x=\frac{-\left(-\frac{13}{25}\right)±\frac{\sqrt{95}}{25}}{2\times \frac{37}{200}}$$

The opposite of -\frac{13}{25} is \frac{13}{25}.

$$x=\frac{\frac{13}{25}±\frac{\sqrt{95}}{25}}{2\times \frac{37}{200}}$$

Multiply 2 times \frac{37}{200}.

$$x=\frac{\frac{13}{25}±\frac{\sqrt{95}}{25}}{\frac{37}{100}}$$

Now solve the equation x=\frac{\frac{13}{25}±\frac{\sqrt{95}}{25}}{\frac{37}{100}} when ± is plus. Add \frac{13}{25} to \frac{\sqrt{95}}{25}.

$$x=\frac{\sqrt{95}+13}{\frac{37}{100}\times 25}$$

Divide \frac{13+\sqrt{95}}{25} by \frac{37}{100} by multiplying \frac{13+\sqrt{95}}{25} by the reciprocal of \frac{37}{100}.

$$x=\frac{4\sqrt{95}+52}{37}$$

Now solve the equation x=\frac{\frac{13}{25}±\frac{\sqrt{95}}{25}}{\frac{37}{100}} when ± is minus. Subtract \frac{\sqrt{95}}{25} from \frac{13}{25}.

$$x=\frac{13-\sqrt{95}}{\frac{37}{100}\times 25}$$

Divide \frac{13-\sqrt{95}}{25} by \frac{37}{100} by multiplying \frac{13-\sqrt{95}}{25} by the reciprocal of \frac{37}{100}.

$$x=\frac{52-4\sqrt{95}}{37}$$

The equation is now solved.

$$x=\frac{4\sqrt{95}+52}{37} x=\frac{52-4\sqrt{95}}{37}$$

Steps for Completing the Square

Reduce the fraction \frac{2}{80} to lowest terms by extracting and canceling out 2.

$$\frac{1}{40}x^{2}+\frac{4}{25}\left(1-x\right)^{2}-0.2x=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

$$\frac{1}{40}x^{2}+\frac{4}{25}\left(1-2x+x^{2}\right)-0.2x=0$$

Use the distributive property to multiply \frac{4}{25} by 1-2x+x^{2}.

$$\frac{1}{40}x^{2}+\frac{4}{25}-\frac{8}{25}x+\frac{4}{25}x^{2}-0.2x=0$$

Combine \frac{1}{40}x^{2} and \frac{4}{25}x^{2} to get \frac{37}{200}x^{2}.

$$\frac{37}{200}x^{2}+\frac{4}{25}-\frac{8}{25}x-0.2x=0$$

Combine -\frac{8}{25}x and -0.2x to get -\frac{13}{25}x.

$$\frac{37}{200}x^{2}+\frac{4}{25}-\frac{13}{25}x=0$$

Subtract \frac{4}{25} from both sides. Anything subtracted from zero gives its negation.

$$\frac{37}{200}x^{2}-\frac{13}{25}x=-\frac{4}{25}$$

Divide both sides of the equation by \frac{37}{200}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{37}{200}x^{2}-\frac{13}{25}x}{\frac{37}{200}}=-\frac{\frac{4}{25}}{\frac{37}{200}}$$

Dividing by \frac{37}{200} undoes the multiplication by \frac{37}{200}.

$$x^{2}+\left(-\frac{\frac{13}{25}}{\frac{37}{200}}\right)x=-\frac{\frac{4}{25}}{\frac{37}{200}}$$

Divide -\frac{13}{25} by \frac{37}{200} by multiplying -\frac{13}{25} by the reciprocal of \frac{37}{200}.

$$x^{2}-\frac{104}{37}x=-\frac{\frac{4}{25}}{\frac{37}{200}}$$

Divide -\frac{4}{25} by \frac{37}{200} by multiplying -\frac{4}{25} by the reciprocal of \frac{37}{200}.

$$x^{2}-\frac{104}{37}x=-\frac{32}{37}$$

Divide -\frac{104}{37}, the coefficient of the x term, by 2 to get -\frac{52}{37}. Then add the square of -\frac{52}{37} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{104}{37}x+\left(-\frac{52}{37}\right)^{2}=-\frac{32}{37}+\left(-\frac{52}{37}\right)^{2}$$

Square -\frac{52}{37} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{104}{37}x+\frac{2704}{1369}=-\frac{32}{37}+\frac{2704}{1369}$$

Add -\frac{32}{37} to \frac{2704}{1369} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{104}{37}x+\frac{2704}{1369}=\frac{1520}{1369}$$

Factor x^{2}-\frac{104}{37}x+\frac{2704}{1369}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{52}{37}\right)^{2}=\frac{1520}{1369}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{52}{37}\right)^{2}}=\sqrt{\frac{1520}{1369}}$$

Simplify.

$$x-\frac{52}{37}=\frac{4\sqrt{95}}{37} x-\frac{52}{37}=-\frac{4\sqrt{95}}{37}$$

Add \frac{52}{37} to both sides of the equation.

$$x=\frac{4\sqrt{95}+52}{37} x=\frac{52-4\sqrt{95}}{37}$$