Solve 4/9x^2-8/3x+5=-1/2x+5

$$\frac { 4 } { 9 } x ^ { 2 } - \frac { 8 } { 3 } x + 5 = - \frac { 1 } { 2 } x + 5$$

Solve for x

$$x = \frac{39}{8} = 4\frac{7}{8} = 4.875$$

$$x=0$$

Steps Using Factoring

Add \frac{1}{2}x to both sides.

$$\frac{4}{9}x^{2}-\frac{8}{3}x+5+\frac{1}{2}x=5$$

Combine -\frac{8}{3}x and \frac{1}{2}x to get -\frac{13}{6}x.

$$\frac{4}{9}x^{2}-\frac{13}{6}x+5=5$$

Subtract 5 from both sides.

$$\frac{4}{9}x^{2}-\frac{13}{6}x+5-5=0$$

Subtract 5 from 5 to get 0.

$$\frac{4}{9}x^{2}-\frac{13}{6}x=0$$

Factor out x.

$$x\left(\frac{4}{9}x-\frac{13}{6}\right)=0$$

To find equation solutions, solve x=0 and \frac{4x}{9}-\frac{13}{6}=0.

$$x=0 x=\frac{39}{8}$$

Steps Using the Quadratic Formula

Add \frac{1}{2}x to both sides.

$$\frac{4}{9}x^{2}-\frac{8}{3}x+5+\frac{1}{2}x=5$$

Combine -\frac{8}{3}x and \frac{1}{2}x to get -\frac{13}{6}x.

$$\frac{4}{9}x^{2}-\frac{13}{6}x+5=5$$

Subtract 5 from both sides.

$$\frac{4}{9}x^{2}-\frac{13}{6}x+5-5=0$$

Subtract 5 from 5 to get 0.

$$\frac{4}{9}x^{2}-\frac{13}{6}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{9} for a, -\frac{13}{6} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{13}{6}\right)±\sqrt{\left(-\frac{13}{6}\right)^{2}}}{2\times \frac{4}{9}}$$

Take the square root of \left(-\frac{13}{6}\right)^{2}.

$$x=\frac{-\left(-\frac{13}{6}\right)±\frac{13}{6}}{2\times \frac{4}{9}}$$

The opposite of -\frac{13}{6} is \frac{13}{6}.

$$x=\frac{\frac{13}{6}±\frac{13}{6}}{2\times \frac{4}{9}}$$

Multiply 2 times \frac{4}{9}.

$$x=\frac{\frac{13}{6}±\frac{13}{6}}{\frac{8}{9}}$$

Now solve the equation x=\frac{\frac{13}{6}±\frac{13}{6}}{\frac{8}{9}} when ± is plus. Add \frac{13}{6} to \frac{13}{6} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{13}{3}}{\frac{8}{9}}$$

Divide \frac{13}{3} by \frac{8}{9} by multiplying \frac{13}{3} by the reciprocal of \frac{8}{9}.

$$x=\frac{39}{8}$$

Now solve the equation x=\frac{\frac{13}{6}±\frac{13}{6}}{\frac{8}{9}} when ± is minus. Subtract \frac{13}{6} from \frac{13}{6} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{\frac{8}{9}}$$

Divide 0 by \frac{8}{9} by multiplying 0 by the reciprocal of \frac{8}{9}.

$$x=0$$

The equation is now solved.

$$x=\frac{39}{8} x=0$$

Steps for Completing the Square

Add \frac{1}{2}x to both sides.

$$\frac{4}{9}x^{2}-\frac{8}{3}x+5+\frac{1}{2}x=5$$

Combine -\frac{8}{3}x and \frac{1}{2}x to get -\frac{13}{6}x.

$$\frac{4}{9}x^{2}-\frac{13}{6}x+5=5$$

Subtract 5 from both sides.

$$\frac{4}{9}x^{2}-\frac{13}{6}x=5-5$$

Subtract 5 from 5 to get 0.

$$\frac{4}{9}x^{2}-\frac{13}{6}x=0$$

Divide both sides of the equation by \frac{4}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{9}x^{2}-\frac{13}{6}x}{\frac{4}{9}}=\frac{0}{\frac{4}{9}}$$

Dividing by \frac{4}{9} undoes the multiplication by \frac{4}{9}.

$$x^{2}+\left(-\frac{\frac{13}{6}}{\frac{4}{9}}\right)x=\frac{0}{\frac{4}{9}}$$

Divide -\frac{13}{6} by \frac{4}{9} by multiplying -\frac{13}{6} by the reciprocal of \frac{4}{9}.

$$x^{2}-\frac{39}{8}x=\frac{0}{\frac{4}{9}}$$

Divide 0 by \frac{4}{9} by multiplying 0 by the reciprocal of \frac{4}{9}.

$$x^{2}-\frac{39}{8}x=0$$

Divide -\frac{39}{8}, the coefficient of the x term, by 2 to get -\frac{39}{16}. Then add the square of -\frac{39}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{39}{8}x+\left(-\frac{39}{16}\right)^{2}=\left(-\frac{39}{16}\right)^{2}$$

Square -\frac{39}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{39}{8}x+\frac{1521}{256}=\frac{1521}{256}$$

Factor x^{2}-\frac{39}{8}x+\frac{1521}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{39}{16}\right)^{2}=\frac{1521}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{16}\right)^{2}}=\sqrt{\frac{1521}{256}}$$

Simplify.

$$x-\frac{39}{16}=\frac{39}{16} x-\frac{39}{16}=-\frac{39}{16}$$

Add \frac{39}{16} to both sides of the equation.

$$x=\frac{39}{8} x=0$$