Solve 21/80x^2+4/25left(1-xright)^2-0.2x=0

$$\frac{ 21 }{ 80 } { x }^{ 2 } + \frac{ 4 }{ 25 } { \left(1-x \right) }^{ 2 } -0.2x = 0$$

Solve for x

$$x=\frac{8}{13}\approx 0.615384615$$

Steps Using the Quadratic Formula

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

$$\frac{21}{80}x^{2}+\frac{4}{25}\left(1-2x+x^{2}\right)-0.2x=0$$

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

$$\frac{21}{80}x^{2}+\frac{4}{25}-\frac{8}{25}x+\frac{4}{25}x^{2}-0.2x=0$$

Combine \frac{21}{80}x^{2} and \frac{4}{25}x^{2} to get \frac{169}{400}x^{2}.

$$\frac{169}{400}x^{2}+\frac{4}{25}-\frac{8}{25}x-0.2x=0$$

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

$$\frac{169}{400}x^{2}+\frac{4}{25}-\frac{13}{25}x=0$$

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.

$$\frac{169}{400}x^{2}-\frac{13}{25}x+\frac{4}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{169}{400} for a, -\frac{13}{25} for b, and \frac{4}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\left(-\frac{13}{25}\right)^{2}-4\times \frac{169}{400}\times \frac{4}{25}}}{2\times \frac{169}{400}}$$

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

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{169}{625}-4\times \frac{169}{400}\times \frac{4}{25}}}{2\times \frac{169}{400}}$$

Multiply -4 times \frac{169}{400}.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{169}{625}-\frac{169}{100}\times \frac{4}{25}}}{2\times \frac{169}{400}}$$

Multiply -\frac{169}{100} times \frac{4}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{\frac{169-169}{625}}}{2\times \frac{169}{400}}$$

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

$$x=\frac{-\left(-\frac{13}{25}\right)±\sqrt{0}}{2\times \frac{169}{400}}$$

Take the square root of 0.

$$x=-\frac{-\frac{13}{25}}{2\times \frac{169}{400}}$$

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

$$x=\frac{\frac{13}{25}}{2\times \frac{169}{400}}$$

Multiply 2 times \frac{169}{400}.

$$x=\frac{\frac{13}{25}}{\frac{169}{200}}$$

Divide \frac{13}{25} by \frac{169}{200} by multiplying \frac{13}{25} by the reciprocal of \frac{169}{200}.

$$x=\frac{8}{13}$$

Steps for Completing the Square

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

$$\frac{21}{80}x^{2}+\frac{4}{25}\left(1-2x+x^{2}\right)-0.2x=0$$

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

$$\frac{21}{80}x^{2}+\frac{4}{25}-\frac{8}{25}x+\frac{4}{25}x^{2}-0.2x=0$$

Combine \frac{21}{80}x^{2} and \frac{4}{25}x^{2} to get \frac{169}{400}x^{2}.

$$\frac{169}{400}x^{2}+\frac{4}{25}-\frac{8}{25}x-0.2x=0$$

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

$$\frac{169}{400}x^{2}+\frac{4}{25}-\frac{13}{25}x=0$$

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

$$\frac{169}{400}x^{2}-\frac{13}{25}x=-\frac{4}{25}$$

Divide both sides of the equation by \frac{169}{400}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{169}{400}x^{2}-\frac{13}{25}x}{\frac{169}{400}}=-\frac{\frac{4}{25}}{\frac{169}{400}}$$

Dividing by \frac{169}{400} undoes the multiplication by \frac{169}{400}.

$$x^{2}+\left(-\frac{\frac{13}{25}}{\frac{169}{400}}\right)x=-\frac{\frac{4}{25}}{\frac{169}{400}}$$

Divide -\frac{13}{25} by \frac{169}{400} by multiplying -\frac{13}{25} by the reciprocal of \frac{169}{400}.

$$x^{2}-\frac{16}{13}x=-\frac{\frac{4}{25}}{\frac{169}{400}}$$

Divide -\frac{4}{25} by \frac{169}{400} by multiplying -\frac{4}{25} by the reciprocal of \frac{169}{400}.

$$x^{2}-\frac{16}{13}x=-\frac{64}{169}$$

Divide -\frac{16}{13}, the coefficient of the x term, by 2 to get -\frac{8}{13}. Then add the square of -\frac{8}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{13}x+\left(-\frac{8}{13}\right)^{2}=-\frac{64}{169}+\left(-\frac{8}{13}\right)^{2}$$

Square -\frac{8}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{13}x+\frac{64}{169}=\frac{-64+64}{169}$$

Add -\frac{64}{169} to \frac{64}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{13}x+\frac{64}{169}=0$$

Factor x^{2}-\frac{16}{13}x+\frac{64}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{13}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{13}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{8}{13}=0 x-\frac{8}{13}=0$$

Add \frac{8}{13} to both sides of the equation.

$$x=\frac{8}{13} x=\frac{8}{13}$$

The equation is now solved. Solutions are the same.

$$x=\frac{8}{13}$$