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

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

Solve for x (complex solution)

$$x=\frac{\sqrt{15}i}{18}+\frac{5}{2}\approx 2.5+0.215165741i$$

$$x=-\frac{\sqrt{15}i}{18}+\frac{5}{2}\approx 2.5-0.215165741i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by x-2.

$$\left(3x-6\right)\left(x-3\right)=-\frac{8}{9}$$

Use the distributive property to multiply 3x-6 by x-3 and combine like terms.

$$3x^{2}-15x+18=-\frac{8}{9}$$

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

$$3x^{2}-15x+18+\frac{8}{9}=0$$

Add 18 and \frac{8}{9} to get \frac{170}{9}.

$$3x^{2}-15x+\frac{170}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -15 for b, and \frac{170}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\times \frac{170}{9}}}{2\times 3}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\times \frac{170}{9}}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\times \frac{170}{9}}}{2\times 3}$$

Multiply -12 times \frac{170}{9}.

$$x=\frac{-\left(-15\right)±\sqrt{225-\frac{680}{3}}}{2\times 3}$$

Add 225 to -\frac{680}{3}.

$$x=\frac{-\left(-15\right)±\sqrt{-\frac{5}{3}}}{2\times 3}$$

Take the square root of -\frac{5}{3}.

$$x=\frac{-\left(-15\right)±\frac{\sqrt{15}i}{3}}{2\times 3}$$

The opposite of -15 is 15.

$$x=\frac{15±\frac{\sqrt{15}i}{3}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{15±\frac{\sqrt{15}i}{3}}{6}$$

Now solve the equation x=\frac{15±\frac{\sqrt{15}i}{3}}{6} when ± is plus. Add 15 to \frac{i\sqrt{15}}{3}.

$$x=\frac{\frac{\sqrt{15}i}{3}+15}{6}$$

Divide 15+\frac{i\sqrt{15}}{3} by 6.

$$x=\frac{\sqrt{15}i}{18}+\frac{5}{2}$$

Now solve the equation x=\frac{15±\frac{\sqrt{15}i}{3}}{6} when ± is minus. Subtract \frac{i\sqrt{15}}{3} from 15.

$$x=\frac{-\frac{\sqrt{15}i}{3}+15}{6}$$

Divide 15-\frac{i\sqrt{15}}{3} by 6.

$$x=-\frac{\sqrt{15}i}{18}+\frac{5}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{15}i}{18}+\frac{5}{2} x=-\frac{\sqrt{15}i}{18}+\frac{5}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3 by x-2.

$$\left(3x-6\right)\left(x-3\right)=-\frac{8}{9}$$

Use the distributive property to multiply 3x-6 by x-3 and combine like terms.

$$3x^{2}-15x+18=-\frac{8}{9}$$

Subtract 18 from both sides.

$$3x^{2}-15x=-\frac{8}{9}-18$$

Subtract 18 from -\frac{8}{9} to get -\frac{170}{9}.

$$3x^{2}-15x=-\frac{170}{9}$$

Divide both sides by 3.

$$\frac{3x^{2}-15x}{3}=-\frac{\frac{170}{9}}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{15}{3}\right)x=-\frac{\frac{170}{9}}{3}$$

Divide -15 by 3.

$$x^{2}-5x=-\frac{\frac{170}{9}}{3}$$

Divide -\frac{170}{9} by 3.

$$x^{2}-5x=-\frac{170}{27}$$

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.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{170}{27}+\left(-\frac{5}{2}\right)^{2}$$

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

$$x^{2}-5x+\frac{25}{4}=-\frac{170}{27}+\frac{25}{4}$$

Add -\frac{170}{27} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5x+\frac{25}{4}=-\frac{5}{108}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{5}{108}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{5}{108}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{15}i}{18} x-\frac{5}{2}=-\frac{\sqrt{15}i}{18}$$

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

$$x=\frac{\sqrt{15}i}{18}+\frac{5}{2} x=-\frac{\sqrt{15}i}{18}+\frac{5}{2}$$