Solve 0=4/3x^2-13/3x

$$0 = \frac { 4 } { 3 } x ^ { 2 } - \frac { 13 } { 3 } x$$

Solve for x

$$x = \frac{13}{4} = 3\frac{1}{4} = 3.25$$

$$x=0$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$\frac{4}{3}x^{2}-\frac{13}{3}x=0$$

Factor out x.

$$x\left(\frac{4}{3}x-\frac{13}{3}\right)=0$$

To find equation solutions, solve x=0 and \frac{4x-13}{3}=0.

$$x=0 x=\frac{13}{4}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$\frac{4}{3}x^{2}-\frac{13}{3}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{3} for a, -\frac{13}{3} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{13}{3}\right)±\sqrt{\left(-\frac{13}{3}\right)^{2}}}{2\times \frac{4}{3}}$$

Take the square root of \left(-\frac{13}{3}\right)^{2}.

$$x=\frac{-\left(-\frac{13}{3}\right)±\frac{13}{3}}{2\times \frac{4}{3}}$$

The opposite of -\frac{13}{3} is \frac{13}{3}.

$$x=\frac{\frac{13}{3}±\frac{13}{3}}{2\times \frac{4}{3}}$$

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

$$x=\frac{\frac{13}{3}±\frac{13}{3}}{\frac{8}{3}}$$

Now solve the equation x=\frac{\frac{13}{3}±\frac{13}{3}}{\frac{8}{3}} when ± is plus. Add \frac{13}{3} to \frac{13}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{26}{3}}{\frac{8}{3}}$$

Divide \frac{26}{3} by \frac{8}{3} by multiplying \frac{26}{3} by the reciprocal of \frac{8}{3}.

$$x=\frac{13}{4}$$

Now solve the equation x=\frac{\frac{13}{3}±\frac{13}{3}}{\frac{8}{3}} when ± is minus. Subtract \frac{13}{3} from \frac{13}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{\frac{8}{3}}$$

Divide 0 by \frac{8}{3} by multiplying 0 by the reciprocal of \frac{8}{3}.

$$x=0$$

The equation is now solved.

$$x=\frac{13}{4} x=0$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$\frac{4}{3}x^{2}-\frac{13}{3}x=0$$

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

$$\frac{\frac{4}{3}x^{2}-\frac{13}{3}x}{\frac{4}{3}}=\frac{0}{\frac{4}{3}}$$

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

$$x^{2}+\left(-\frac{\frac{13}{3}}{\frac{4}{3}}\right)x=\frac{0}{\frac{4}{3}}$$

Divide -\frac{13}{3} by \frac{4}{3} by multiplying -\frac{13}{3} by the reciprocal of \frac{4}{3}.

$$x^{2}-\frac{13}{4}x=\frac{0}{\frac{4}{3}}$$

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

$$x^{2}-\frac{13}{4}x=0$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{169}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{13}{8} x-\frac{13}{8}=-\frac{13}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=\frac{13}{4} x=0$$