Solve x^2(6%)^2+(1-x)^2(2%)^2+2x(1-x)*0.12*6%*2%=0.0327

$$x ^ { 2 } ( 6 \% ) ^ { 2 } + ( 1 - x ) ^ { 2 } ( 2 \% ) ^ { 2 } + 2 x ( 1 - x ) \times 0.12 \times 6 \% \times 2 \% = 0.0327$$

Solve for x

$$x=\frac{\sqrt{468606}}{232}+\frac{2}{29}\approx 3.019603561$$

$$x=-\frac{\sqrt{468606}}{232}+\frac{2}{29}\approx -2.881672526$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{6}{100} to lowest terms by extracting and canceling out 2.

$$x^{2}\times \left(\frac{3}{50}\right)^{2}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Calculate \frac{3}{50} to the power of 2 and get \frac{9}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

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

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Reduce the fraction \frac{2}{100} to lowest terms by extracting and canceling out 2.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{1}{50}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Calculate \frac{1}{50} to the power of 2 and get \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \frac{1}{2500}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Use the distributive property to multiply 1-2x+x^{2} by \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\frac{1}{2500}-\frac{1}{1250}x+\frac{1}{2500}x^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Combine x^{2}\times \frac{9}{2500} and \frac{1}{2500}x^{2} to get \frac{1}{250}x^{2}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Multiply 2 and 0.12 to get 0.24.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0.24x\left(1-x\right)\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Reduce the fraction \frac{6}{100} to lowest terms by extracting and canceling out 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0.24x\left(1-x\right)\times \frac{3}{50}\times \frac{2}{100}=0.0327$$

Multiply 0.24 and \frac{3}{50} to get \frac{9}{625}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{625}x\left(1-x\right)\times \frac{2}{100}=0.0327$$

Reduce the fraction \frac{2}{100} to lowest terms by extracting and canceling out 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{625}x\left(1-x\right)\times \frac{1}{50}=0.0327$$

Multiply \frac{9}{625} and \frac{1}{50} to get \frac{9}{31250}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{31250}x\left(1-x\right)=0.0327$$

Use the distributive property to multiply \frac{9}{31250}x by 1-x.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{31250}x-\frac{9}{31250}x^{2}=0.0327$$

Combine -\frac{1}{1250}x and \frac{9}{31250}x to get -\frac{8}{15625}x.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{8}{15625}x-\frac{9}{31250}x^{2}=0.0327$$

Combine \frac{1}{250}x^{2} and -\frac{9}{31250}x^{2} to get \frac{58}{15625}x^{2}.

$$\frac{58}{15625}x^{2}+\frac{1}{2500}-\frac{8}{15625}x=0.0327$$

Subtract 0.0327 from both sides.

$$\frac{58}{15625}x^{2}+\frac{1}{2500}-\frac{8}{15625}x-0.0327=0$$

Subtract 0.0327 from \frac{1}{2500} to get -\frac{323}{10000}.

$$\frac{58}{15625}x^{2}-\frac{323}{10000}-\frac{8}{15625}x=0$$

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.

$$\frac{58}{15625}x^{2}-\frac{8}{15625}x-\frac{323}{10000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{58}{15625} for a, -\frac{8}{15625} for b, and -\frac{323}{10000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{8}{15625}\right)±\sqrt{\left(-\frac{8}{15625}\right)^{2}-4\times \frac{58}{15625}\left(-\frac{323}{10000}\right)}}{2\times \frac{58}{15625}}$$

Square -\frac{8}{15625} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{8}{15625}\right)±\sqrt{\frac{64}{244140625}-4\times \frac{58}{15625}\left(-\frac{323}{10000}\right)}}{2\times \frac{58}{15625}}$$

Multiply -4 times \frac{58}{15625}.

$$x=\frac{-\left(-\frac{8}{15625}\right)±\sqrt{\frac{64}{244140625}-\frac{232}{15625}\left(-\frac{323}{10000}\right)}}{2\times \frac{58}{15625}}$$

Multiply -\frac{232}{15625} times -\frac{323}{10000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{8}{15625}\right)±\sqrt{\frac{64}{244140625}+\frac{9367}{19531250}}}{2\times \frac{58}{15625}}$$

Add \frac{64}{244140625} to \frac{9367}{19531250} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{8}{15625}\right)±\sqrt{\frac{234303}{488281250}}}{2\times \frac{58}{15625}}$$

Take the square root of \frac{234303}{488281250}.

$$x=\frac{-\left(-\frac{8}{15625}\right)±\frac{\sqrt{468606}}{31250}}{2\times \frac{58}{15625}}$$

The opposite of -\frac{8}{15625} is \frac{8}{15625}.

$$x=\frac{\frac{8}{15625}±\frac{\sqrt{468606}}{31250}}{2\times \frac{58}{15625}}$$

Multiply 2 times \frac{58}{15625}.

$$x=\frac{\frac{8}{15625}±\frac{\sqrt{468606}}{31250}}{\frac{116}{15625}}$$

Now solve the equation x=\frac{\frac{8}{15625}±\frac{\sqrt{468606}}{31250}}{\frac{116}{15625}} when ± is plus. Add \frac{8}{15625} to \frac{\sqrt{468606}}{31250}.

$$x=\frac{\frac{\sqrt{468606}}{31250}+\frac{8}{15625}}{\frac{116}{15625}}$$

Divide \frac{8}{15625}+\frac{\sqrt{468606}}{31250} by \frac{116}{15625} by multiplying \frac{8}{15625}+\frac{\sqrt{468606}}{31250} by the reciprocal of \frac{116}{15625}.

$$x=\frac{\sqrt{468606}}{232}+\frac{2}{29}$$

Now solve the equation x=\frac{\frac{8}{15625}±\frac{\sqrt{468606}}{31250}}{\frac{116}{15625}} when ± is minus. Subtract \frac{\sqrt{468606}}{31250} from \frac{8}{15625}.

$$x=\frac{-\frac{\sqrt{468606}}{31250}+\frac{8}{15625}}{\frac{116}{15625}}$$

Divide \frac{8}{15625}-\frac{\sqrt{468606}}{31250} by \frac{116}{15625} by multiplying \frac{8}{15625}-\frac{\sqrt{468606}}{31250} by the reciprocal of \frac{116}{15625}.

$$x=-\frac{\sqrt{468606}}{232}+\frac{2}{29}$$

The equation is now solved.

$$x=\frac{\sqrt{468606}}{232}+\frac{2}{29} x=-\frac{\sqrt{468606}}{232}+\frac{2}{29}$$

Steps for Completing the Square

Reduce the fraction \frac{6}{100} to lowest terms by extracting and canceling out 2.

$$x^{2}\times \left(\frac{3}{50}\right)^{2}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Calculate \frac{3}{50} to the power of 2 and get \frac{9}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

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

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Reduce the fraction \frac{2}{100} to lowest terms by extracting and canceling out 2.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{1}{50}\right)^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Calculate \frac{1}{50} to the power of 2 and get \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \frac{1}{2500}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Use the distributive property to multiply 1-2x+x^{2} by \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\frac{1}{2500}-\frac{1}{1250}x+\frac{1}{2500}x^{2}+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Combine x^{2}\times \frac{9}{2500} and \frac{1}{2500}x^{2} to get \frac{1}{250}x^{2}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+2x\left(1-x\right)\times 0.12\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Multiply 2 and 0.12 to get 0.24.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0.24x\left(1-x\right)\times \frac{6}{100}\times \frac{2}{100}=0.0327$$

Reduce the fraction \frac{6}{100} to lowest terms by extracting and canceling out 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0.24x\left(1-x\right)\times \frac{3}{50}\times \frac{2}{100}=0.0327$$

Multiply 0.24 and \frac{3}{50} to get \frac{9}{625}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{625}x\left(1-x\right)\times \frac{2}{100}=0.0327$$

Reduce the fraction \frac{2}{100} to lowest terms by extracting and canceling out 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{625}x\left(1-x\right)\times \frac{1}{50}=0.0327$$

Multiply \frac{9}{625} and \frac{1}{50} to get \frac{9}{31250}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{31250}x\left(1-x\right)=0.0327$$

Use the distributive property to multiply \frac{9}{31250}x by 1-x.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+\frac{9}{31250}x-\frac{9}{31250}x^{2}=0.0327$$

Combine -\frac{1}{1250}x and \frac{9}{31250}x to get -\frac{8}{15625}x.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{8}{15625}x-\frac{9}{31250}x^{2}=0.0327$$

Combine \frac{1}{250}x^{2} and -\frac{9}{31250}x^{2} to get \frac{58}{15625}x^{2}.

$$\frac{58}{15625}x^{2}+\frac{1}{2500}-\frac{8}{15625}x=0.0327$$

Subtract \frac{1}{2500} from both sides.

$$\frac{58}{15625}x^{2}-\frac{8}{15625}x=0.0327-\frac{1}{2500}$$

Subtract \frac{1}{2500} from 0.0327 to get \frac{323}{10000}.

$$\frac{58}{15625}x^{2}-\frac{8}{15625}x=\frac{323}{10000}$$

Divide both sides of the equation by \frac{58}{15625}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{58}{15625}x^{2}-\frac{8}{15625}x}{\frac{58}{15625}}=\frac{\frac{323}{10000}}{\frac{58}{15625}}$$

Dividing by \frac{58}{15625} undoes the multiplication by \frac{58}{15625}.

$$x^{2}+\left(-\frac{\frac{8}{15625}}{\frac{58}{15625}}\right)x=\frac{\frac{323}{10000}}{\frac{58}{15625}}$$

Divide -\frac{8}{15625} by \frac{58}{15625} by multiplying -\frac{8}{15625} by the reciprocal of \frac{58}{15625}.

$$x^{2}-\frac{4}{29}x=\frac{\frac{323}{10000}}{\frac{58}{15625}}$$

Divide \frac{323}{10000} by \frac{58}{15625} by multiplying \frac{323}{10000} by the reciprocal of \frac{58}{15625}.

$$x^{2}-\frac{4}{29}x=\frac{8075}{928}$$

Divide -\frac{4}{29}, the coefficient of the x term, by 2 to get -\frac{2}{29}. Then add the square of -\frac{2}{29} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{29}x+\left(-\frac{2}{29}\right)^{2}=\frac{8075}{928}+\left(-\frac{2}{29}\right)^{2}$$

Square -\frac{2}{29} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{29}x+\frac{4}{841}=\frac{8075}{928}+\frac{4}{841}$$

Add \frac{8075}{928} to \frac{4}{841} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{29}x+\frac{4}{841}=\frac{234303}{26912}$$

Factor x^{2}-\frac{4}{29}x+\frac{4}{841}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{29}\right)^{2}=\frac{234303}{26912}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{29}\right)^{2}}=\sqrt{\frac{234303}{26912}}$$

Simplify.

$$x-\frac{2}{29}=\frac{\sqrt{468606}}{232} x-\frac{2}{29}=-\frac{\sqrt{468606}}{232}$$

Add \frac{2}{29} to both sides of the equation.

$$x=\frac{\sqrt{468606}}{232}+\frac{2}{29} x=-\frac{\sqrt{468606}}{232}+\frac{2}{29}$$