Solve -3/4x^2+2x+1/2x=0

$$- \frac { 3 } { 4 } x ^ { 2 } + 2 x + \frac { 1 } { 2 } x = 0$$

Solve for x

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(-\frac{3}{4}x+2+\frac{1}{2}\right)=0$$

To find equation solutions, solve x=0 and -\frac{3x}{4}+\frac{5}{2}=0.

$$x=0 x=\frac{10}{3}$$

Steps Using the Quadratic Formula

Combine 2x and \frac{1}{2}x to get \frac{5}{2}x.

$$-\frac{3}{4}x^{2}+\frac{5}{2}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{3}{4} for a, \frac{5}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{5}{2}±\sqrt{\left(\frac{5}{2}\right)^{2}}}{2\left(-\frac{3}{4}\right)}$$

Take the square root of \left(\frac{5}{2}\right)^{2}.

$$x=\frac{-\frac{5}{2}±\frac{5}{2}}{2\left(-\frac{3}{4}\right)}$$

Multiply 2 times -\frac{3}{4}.

$$x=\frac{-\frac{5}{2}±\frac{5}{2}}{-\frac{3}{2}}$$

Now solve the equation x=\frac{-\frac{5}{2}±\frac{5}{2}}{-\frac{3}{2}} when ± is plus. Add -\frac{5}{2} to \frac{5}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-\frac{3}{2}}$$

Divide 0 by -\frac{3}{2} by multiplying 0 by the reciprocal of -\frac{3}{2}.

$$x=0$$

Now solve the equation x=\frac{-\frac{5}{2}±\frac{5}{2}}{-\frac{3}{2}} when ± is minus. Subtract \frac{5}{2} from -\frac{5}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{5}{-\frac{3}{2}}$$

Divide -5 by -\frac{3}{2} by multiplying -5 by the reciprocal of -\frac{3}{2}.

$$x=\frac{10}{3}$$

The equation is now solved.

$$x=0 x=\frac{10}{3}$$

Steps for Completing the Square

Combine 2x and \frac{1}{2}x to get \frac{5}{2}x.

$$-\frac{3}{4}x^{2}+\frac{5}{2}x=0$$

Divide both sides of the equation by -\frac{3}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{3}{4}x^{2}+\frac{5}{2}x}{-\frac{3}{4}}=\frac{0}{-\frac{3}{4}}$$

Dividing by -\frac{3}{4} undoes the multiplication by -\frac{3}{4}.

$$x^{2}+\frac{\frac{5}{2}}{-\frac{3}{4}}x=\frac{0}{-\frac{3}{4}}$$

Divide \frac{5}{2} by -\frac{3}{4} by multiplying \frac{5}{2} by the reciprocal of -\frac{3}{4}.

$$x^{2}-\frac{10}{3}x=\frac{0}{-\frac{3}{4}}$$

Divide 0 by -\frac{3}{4} by multiplying 0 by the reciprocal of -\frac{3}{4}.

$$x^{2}-\frac{10}{3}x=0$$

Divide -\frac{10}{3}, the coefficient of the x term, by 2 to get -\frac{5}{3}. Then add the square of -\frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{3}x+\left(-\frac{5}{3}\right)^{2}=\left(-\frac{5}{3}\right)^{2}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=\frac{25}{9}$$

Factor x^{2}-\frac{10}{3}x+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{3}\right)^{2}=\frac{25}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{3}\right)^{2}}=\sqrt{\frac{25}{9}}$$

Simplify.

$$x-\frac{5}{3}=\frac{5}{3} x-\frac{5}{3}=-\frac{5}{3}$$

Add \frac{5}{3} to both sides of the equation.

$$x=\frac{10}{3} x=0$$