Solve (0.5-x)(100-50x)=(-75-50x)x

$$( 0.5 - x ) ( 100 - 50 x ) = ( - 75 - 50 x ) x$$

Solve for x (complex solution)

$$x=\frac{1+i\sqrt{7}}{4}\approx 0.25+0.661437828i$$

$$x=\frac{-i\sqrt{7}+1}{4}\approx 0.25-0.661437828i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.5-x by 100-50x and combine like terms.

$$50-125x+50x^{2}=\left(-75-50x\right)x$$

Use the distributive property to multiply -75-50x by x.

$$50-125x+50x^{2}=-75x-50x^{2}$$

Add 75x to both sides.

$$50-125x+50x^{2}+75x=-50x^{2}$$

Combine -125x and 75x to get -50x.

$$50-50x+50x^{2}=-50x^{2}$$

Add 50x^{2} to both sides.

$$50-50x+50x^{2}+50x^{2}=0$$

Combine 50x^{2} and 50x^{2} to get 100x^{2}.

$$50-50x+100x^{2}=0$$

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.

$$100x^{2}-50x+50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, -50 for b, and 50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 100\times 50}}{2\times 100}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 100\times 50}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-\left(-50\right)±\sqrt{2500-400\times 50}}{2\times 100}$$

Multiply -400 times 50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-20000}}{2\times 100}$$

Add 2500 to -20000.

$$x=\frac{-\left(-50\right)±\sqrt{-17500}}{2\times 100}$$

Take the square root of -17500.

$$x=\frac{-\left(-50\right)±50\sqrt{7}i}{2\times 100}$$

The opposite of -50 is 50.

$$x=\frac{50±50\sqrt{7}i}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{50±50\sqrt{7}i}{200}$$

Now solve the equation x=\frac{50±50\sqrt{7}i}{200} when ± is plus. Add 50 to 50i\sqrt{7}.

$$x=\frac{50+50\sqrt{7}i}{200}$$

Divide 50+50i\sqrt{7} by 200.

$$x=\frac{1+\sqrt{7}i}{4}$$

Now solve the equation x=\frac{50±50\sqrt{7}i}{200} when ± is minus. Subtract 50i\sqrt{7} from 50.

$$x=\frac{-50\sqrt{7}i+50}{200}$$

Divide 50-50i\sqrt{7} by 200.

$$x=\frac{-\sqrt{7}i+1}{4}$$

The equation is now solved.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 0.5-x by 100-50x and combine like terms.

$$50-125x+50x^{2}=\left(-75-50x\right)x$$

Use the distributive property to multiply -75-50x by x.

$$50-125x+50x^{2}=-75x-50x^{2}$$

Add 75x to both sides.

$$50-125x+50x^{2}+75x=-50x^{2}$$

Combine -125x and 75x to get -50x.

$$50-50x+50x^{2}=-50x^{2}$$

Add 50x^{2} to both sides.

$$50-50x+50x^{2}+50x^{2}=0$$

Combine 50x^{2} and 50x^{2} to get 100x^{2}.

$$50-50x+100x^{2}=0$$

Subtract 50 from both sides. Anything subtracted from zero gives its negation.

$$-50x+100x^{2}=-50$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$100x^{2}-50x=-50$$

Divide both sides by 100.

$$\frac{100x^{2}-50x}{100}=-\frac{50}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\left(-\frac{50}{100}\right)x=-\frac{50}{100}$$

Reduce the fraction \frac{-50}{100} to lowest terms by extracting and canceling out 50.

$$x^{2}-\frac{1}{2}x=-\frac{50}{100}$$

Reduce the fraction \frac{-50}{100} to lowest terms by extracting and canceling out 50.

$$x^{2}-\frac{1}{2}x=-\frac{1}{2}$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{1}{2}+\frac{1}{16}$$

Add -\frac{1}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{7}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{7}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{\sqrt{7}i}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$