Solve 0.25^2x^2+0.4^2(1-x)^2+0.2*(1-x)=0

$$0.25 ^ { 2 } x ^ { 2 } + 0.4 ^ { 2 } ( 1 - x ) ^ { 2 } + 0.2 \times ( 1 - x ) = 0$$

Solve for x (complex solution)

$$x=\frac{104+i\times 20\sqrt{5}}{89}\approx 1.168539326+0.502487186i$$

$$x=\frac{-i\times 20\sqrt{5}+104}{89}\approx 1.168539326-0.502487186i$$

Steps Using the Quadratic Formula

Calculate 0.25 to the power of 2 and get 0.0625.

$$0.0625x^{2}+0.4^{2}\left(1-x\right)^{2}+0.2\left(1-x\right)=0$$

Calculate 0.4 to the power of 2 and get 0.16.

$$0.0625x^{2}+0.16\left(1-x\right)^{2}+0.2\left(1-x\right)=0$$

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

$$0.0625x^{2}+0.16\left(1-2x+x^{2}\right)+0.2\left(1-x\right)=0$$

Use the distributive property to multiply 0.16 by 1-2x+x^{2}.

$$0.0625x^{2}+0.16-0.32x+0.16x^{2}+0.2\left(1-x\right)=0$$

Combine 0.0625x^{2} and 0.16x^{2} to get 0.2225x^{2}.

$$0.2225x^{2}+0.16-0.32x+0.2\left(1-x\right)=0$$

Use the distributive property to multiply 0.2 by 1-x.

$$0.2225x^{2}+0.16-0.32x+0.2-0.2x=0$$

Add 0.16 and 0.2 to get 0.36.

$$0.2225x^{2}+0.36-0.32x-0.2x=0$$

Combine -0.32x and -0.2x to get -0.52x.

$$0.2225x^{2}+0.36-0.52x=0$$

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.

$$0.2225x^{2}-0.52x+0.36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.2225 for a, -0.52 for b, and 0.36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.52\right)±\sqrt{\left(-0.52\right)^{2}-4\times 0.2225\times 0.36}}{2\times 0.2225}$$

Square -0.52 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.52\right)±\sqrt{0.2704-4\times 0.2225\times 0.36}}{2\times 0.2225}$$

Multiply -4 times 0.2225.

$$x=\frac{-\left(-0.52\right)±\sqrt{0.2704-0.89\times 0.36}}{2\times 0.2225}$$

Multiply -0.89 times 0.36 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.52\right)±\sqrt{0.2704-0.3204}}{2\times 0.2225}$$

Add 0.2704 to -0.3204 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.52\right)±\sqrt{-0.05}}{2\times 0.2225}$$

Take the square root of -0.05.

$$x=\frac{-\left(-0.52\right)±\frac{\sqrt{5}i}{10}}{2\times 0.2225}$$

The opposite of -0.52 is 0.52.

$$x=\frac{0.52±\frac{\sqrt{5}i}{10}}{2\times 0.2225}$$

Multiply 2 times 0.2225.

$$x=\frac{0.52±\frac{\sqrt{5}i}{10}}{0.445}$$

Now solve the equation x=\frac{0.52±\frac{\sqrt{5}i}{10}}{0.445} when ± is plus. Add 0.52 to \frac{i\sqrt{5}}{10}.

$$x=\frac{\frac{\sqrt{5}i}{10}+\frac{13}{25}}{0.445}$$

Divide \frac{13}{25}+\frac{i\sqrt{5}}{10} by 0.445 by multiplying \frac{13}{25}+\frac{i\sqrt{5}}{10} by the reciprocal of 0.445.

$$x=\frac{104+20\sqrt{5}i}{89}$$

Now solve the equation x=\frac{0.52±\frac{\sqrt{5}i}{10}}{0.445} when ± is minus. Subtract \frac{i\sqrt{5}}{10} from 0.52.

$$x=\frac{-\frac{\sqrt{5}i}{10}+\frac{13}{25}}{0.445}$$

Divide \frac{13}{25}-\frac{i\sqrt{5}}{10} by 0.445 by multiplying \frac{13}{25}-\frac{i\sqrt{5}}{10} by the reciprocal of 0.445.

$$x=\frac{-20\sqrt{5}i+104}{89}$$

The equation is now solved.

$$x=\frac{104+20\sqrt{5}i}{89} x=\frac{-20\sqrt{5}i+104}{89}$$

Steps for Completing the Square

Calculate 0.25 to the power of 2 and get 0.0625.

$$0.0625x^{2}+0.4^{2}\left(1-x\right)^{2}+0.2\left(1-x\right)=0$$

Calculate 0.4 to the power of 2 and get 0.16.

$$0.0625x^{2}+0.16\left(1-x\right)^{2}+0.2\left(1-x\right)=0$$

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

$$0.0625x^{2}+0.16\left(1-2x+x^{2}\right)+0.2\left(1-x\right)=0$$

Use the distributive property to multiply 0.16 by 1-2x+x^{2}.

$$0.0625x^{2}+0.16-0.32x+0.16x^{2}+0.2\left(1-x\right)=0$$

Combine 0.0625x^{2} and 0.16x^{2} to get 0.2225x^{2}.

$$0.2225x^{2}+0.16-0.32x+0.2\left(1-x\right)=0$$

Use the distributive property to multiply 0.2 by 1-x.

$$0.2225x^{2}+0.16-0.32x+0.2-0.2x=0$$

Add 0.16 and 0.2 to get 0.36.

$$0.2225x^{2}+0.36-0.32x-0.2x=0$$

Combine -0.32x and -0.2x to get -0.52x.

$$0.2225x^{2}+0.36-0.52x=0$$

Subtract 0.36 from both sides. Anything subtracted from zero gives its negation.

$$0.2225x^{2}-0.52x=-0.36$$

Divide both sides of the equation by 0.2225, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.2225x^{2}-0.52x}{0.2225}=-\frac{0.36}{0.2225}$$

Dividing by 0.2225 undoes the multiplication by 0.2225.

$$x^{2}+\left(-\frac{0.52}{0.2225}\right)x=-\frac{0.36}{0.2225}$$

Divide -0.52 by 0.2225 by multiplying -0.52 by the reciprocal of 0.2225.

$$x^{2}-\frac{208}{89}x=-\frac{0.36}{0.2225}$$

Divide -0.36 by 0.2225 by multiplying -0.36 by the reciprocal of 0.2225.

$$x^{2}-\frac{208}{89}x=-\frac{144}{89}$$

Divide -\frac{208}{89}, the coefficient of the x term, by 2 to get -\frac{104}{89}. Then add the square of -\frac{104}{89} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{208}{89}x+\left(-\frac{104}{89}\right)^{2}=-\frac{144}{89}+\left(-\frac{104}{89}\right)^{2}$$

Square -\frac{104}{89} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{208}{89}x+\frac{10816}{7921}=-\frac{144}{89}+\frac{10816}{7921}$$

Add -\frac{144}{89} to \frac{10816}{7921} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{208}{89}x+\frac{10816}{7921}=-\frac{2000}{7921}$$

Factor x^{2}-\frac{208}{89}x+\frac{10816}{7921}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{104}{89}\right)^{2}=-\frac{2000}{7921}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{104}{89}\right)^{2}}=\sqrt{-\frac{2000}{7921}}$$

Simplify.

$$x-\frac{104}{89}=\frac{20\sqrt{5}i}{89} x-\frac{104}{89}=-\frac{20\sqrt{5}i}{89}$$

Add \frac{104}{89} to both sides of the equation.

$$x=\frac{104+20\sqrt{5}i}{89} x=\frac{-20\sqrt{5}i+104}{89}$$