Solve -22.9x+(-6.1*(x-2)*(x/2))=0

$$-22.9x+(-6.1 \times (x-2) \times ( \frac{ x }{ 2 } ))=0$$

Solve for x

$$x = -\frac{336}{61} = -5\frac{31}{61} \approx -5.508196721$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$-45.8x-6.1\left(x-2\right)x=0$$

Use the distributive property to multiply -6.1 by x-2.

$$-45.8x+\left(-6.1x+12.2\right)x=0$$

Use the distributive property to multiply -6.1x+12.2 by x.

$$-45.8x-6.1x^{2}+12.2x=0$$

Combine -45.8x and 12.2x to get -33.6x.

$$-33.6x-6.1x^{2}=0$$

Factor out x.

$$x\left(-33.6-6.1x\right)=0$$

To find equation solutions, solve x=0 and -33.6-\frac{61x}{10}=0.

$$x=0 x=-\frac{336}{61}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$-45.8x-6.1\left(x-2\right)x=0$$

Use the distributive property to multiply -6.1 by x-2.

$$-45.8x+\left(-6.1x+12.2\right)x=0$$

Use the distributive property to multiply -6.1x+12.2 by x.

$$-45.8x-6.1x^{2}+12.2x=0$$

Combine -45.8x and 12.2x to get -33.6x.

$$-33.6x-6.1x^{2}=0$$

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.

$$-6.1x^{2}-33.6x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6.1 for a, -33.6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33.6\right)±\sqrt{\left(-33.6\right)^{2}}}{2\left(-6.1\right)}$$

Take the square root of \left(-33.6\right)^{2}.

$$x=\frac{-\left(-33.6\right)±\frac{168}{5}}{2\left(-6.1\right)}$$

The opposite of -33.6 is 33.6.

$$x=\frac{33.6±\frac{168}{5}}{2\left(-6.1\right)}$$

Multiply 2 times -6.1.

$$x=\frac{33.6±\frac{168}{5}}{-12.2}$$

Now solve the equation x=\frac{33.6±\frac{168}{5}}{-12.2} when ± is plus. Add 33.6 to \frac{168}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{336}{5}}{-12.2}$$

Divide \frac{336}{5} by -12.2 by multiplying \frac{336}{5} by the reciprocal of -12.2.

$$x=-\frac{336}{61}$$

Now solve the equation x=\frac{33.6±\frac{168}{5}}{-12.2} when ± is minus. Subtract \frac{168}{5} from 33.6 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-12.2}$$

Divide 0 by -12.2 by multiplying 0 by the reciprocal of -12.2.

$$x=0$$

The equation is now solved.

$$x=-\frac{336}{61} x=0$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$-45.8x-6.1\left(x-2\right)x=0$$

Use the distributive property to multiply -6.1 by x-2.

$$-45.8x+\left(-6.1x+12.2\right)x=0$$

Use the distributive property to multiply -6.1x+12.2 by x.

$$-45.8x-6.1x^{2}+12.2x=0$$

Combine -45.8x and 12.2x to get -33.6x.

$$-33.6x-6.1x^{2}=0$$

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.

$$-6.1x^{2}-33.6x=0$$

Divide both sides of the equation by -6.1, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-6.1x^{2}-33.6x}{-6.1}=\frac{0}{-6.1}$$

Dividing by -6.1 undoes the multiplication by -6.1.

$$x^{2}+\left(-\frac{33.6}{-6.1}\right)x=\frac{0}{-6.1}$$

Divide -33.6 by -6.1 by multiplying -33.6 by the reciprocal of -6.1.

$$x^{2}+\frac{336}{61}x=\frac{0}{-6.1}$$

Divide 0 by -6.1 by multiplying 0 by the reciprocal of -6.1.

$$x^{2}+\frac{336}{61}x=0$$

Divide \frac{336}{61}, the coefficient of the x term, by 2 to get \frac{168}{61}. Then add the square of \frac{168}{61} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{336}{61}x+\frac{168}{61}^{2}=\frac{168}{61}^{2}$$

Square \frac{168}{61} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{336}{61}x+\frac{28224}{3721}=\frac{28224}{3721}$$

Factor x^{2}+\frac{336}{61}x+\frac{28224}{3721}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{168}{61}\right)^{2}=\frac{28224}{3721}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{168}{61}\right)^{2}}=\sqrt{\frac{28224}{3721}}$$

Simplify.

$$x+\frac{168}{61}=\frac{168}{61} x+\frac{168}{61}=-\frac{168}{61}$$

Subtract \frac{168}{61} from both sides of the equation.

$$x=0 x=-\frac{336}{61}$$