Solve -8/9=4(x-2)(x-3)

$$- \frac { 8 } { 9 } = 4 ( x - 2 ) ( x - 3 )$$

Solve for x

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by x-2.

$$-\frac{8}{9}=\left(4x-8\right)\left(x-3\right)$$

Use the distributive property to multiply 4x-8 by x-3 and combine like terms.

$$-\frac{8}{9}=4x^{2}-20x+24$$

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

$$4x^{2}-20x+24=-\frac{8}{9}$$

Add \frac{8}{9} to both sides.

$$4x^{2}-20x+24+\frac{8}{9}=0$$

Add 24 and \frac{8}{9} to get \frac{224}{9}.

$$4x^{2}-20x+\frac{224}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -20 for b, and \frac{224}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 4\times \frac{224}{9}}}{2\times 4}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 4\times \frac{224}{9}}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-20\right)±\sqrt{400-16\times \frac{224}{9}}}{2\times 4}$$

Multiply -16 times \frac{224}{9}.

$$x=\frac{-\left(-20\right)±\sqrt{400-\frac{3584}{9}}}{2\times 4}$$

Add 400 to -\frac{3584}{9}.

$$x=\frac{-\left(-20\right)±\sqrt{\frac{16}{9}}}{2\times 4}$$

Take the square root of \frac{16}{9}.

$$x=\frac{-\left(-20\right)±\frac{4}{3}}{2\times 4}$$

The opposite of -20 is 20.

$$x=\frac{20±\frac{4}{3}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{20±\frac{4}{3}}{8}$$

Now solve the equation x=\frac{20±\frac{4}{3}}{8} when ± is plus. Add 20 to \frac{4}{3}.

$$x=\frac{\frac{64}{3}}{8}$$

Divide \frac{64}{3} by 8.

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

Now solve the equation x=\frac{20±\frac{4}{3}}{8} when ± is minus. Subtract \frac{4}{3} from 20.

$$x=\frac{\frac{56}{3}}{8}$$

Divide \frac{56}{3} by 8.

$$x=\frac{7}{3}$$

The equation is now solved.

$$x=\frac{8}{3} x=\frac{7}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 4 by x-2.

$$-\frac{8}{9}=\left(4x-8\right)\left(x-3\right)$$

Use the distributive property to multiply 4x-8 by x-3 and combine like terms.

$$-\frac{8}{9}=4x^{2}-20x+24$$

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

$$4x^{2}-20x+24=-\frac{8}{9}$$

Subtract 24 from both sides.

$$4x^{2}-20x=-\frac{8}{9}-24$$

Subtract 24 from -\frac{8}{9} to get -\frac{224}{9}.

$$4x^{2}-20x=-\frac{224}{9}$$

Divide both sides by 4.

$$\frac{4x^{2}-20x}{4}=-\frac{\frac{224}{9}}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{20}{4}\right)x=-\frac{\frac{224}{9}}{4}$$

Divide -20 by 4.

$$x^{2}-5x=-\frac{\frac{224}{9}}{4}$$

Divide -\frac{224}{9} by 4.

$$x^{2}-5x=-\frac{56}{9}$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{56}{9}+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-\frac{56}{9}+\frac{25}{4}$$

Add -\frac{56}{9} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5x+\frac{25}{4}=\frac{1}{36}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$x-\frac{5}{2}=\frac{1}{6} x-\frac{5}{2}=-\frac{1}{6}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{8}{3} x=\frac{7}{3}$$