Solve (x-1)(x-2/3)=18

$$( x - 1 ) ( x - \frac { 2 } { 3 } ) = 18$$

Solve for x

$$x = \frac{\sqrt{649} + 5}{6} \approx 5.079246401$$

$$x=\frac{5-\sqrt{649}}{6}\approx -3.412579734$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-1 by x-\frac{2}{3} and combine like terms.

$$x^{2}-\frac{5}{3}x+\frac{2}{3}=18$$

Subtract 18 from both sides.

$$x^{2}-\frac{5}{3}x+\frac{2}{3}-18=0$$

Subtract 18 from \frac{2}{3} to get -\frac{52}{3}.

$$x^{2}-\frac{5}{3}x-\frac{52}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{5}{3} for b, and -\frac{52}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\left(-\frac{5}{3}\right)^{2}-4\left(-\frac{52}{3}\right)}}{2}$$

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

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{25}{9}-4\left(-\frac{52}{3}\right)}}{2}$$

Multiply -4 times -\frac{52}{3}.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{25}{9}+\frac{208}{3}}}{2}$$

Add \frac{25}{9} to \frac{208}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{649}{9}}}{2}$$

Take the square root of \frac{649}{9}.

$$x=\frac{-\left(-\frac{5}{3}\right)±\frac{\sqrt{649}}{3}}{2}$$

The opposite of -\frac{5}{3} is \frac{5}{3}.

$$x=\frac{\frac{5}{3}±\frac{\sqrt{649}}{3}}{2}$$

Now solve the equation x=\frac{\frac{5}{3}±\frac{\sqrt{649}}{3}}{2} when ± is plus. Add \frac{5}{3} to \frac{\sqrt{649}}{3}.

$$x=\frac{\sqrt{649}+5}{2\times 3}$$

Divide \frac{5+\sqrt{649}}{3} by 2.

$$x=\frac{\sqrt{649}+5}{6}$$

Now solve the equation x=\frac{\frac{5}{3}±\frac{\sqrt{649}}{3}}{2} when ± is minus. Subtract \frac{\sqrt{649}}{3} from \frac{5}{3}.

$$x=\frac{5-\sqrt{649}}{2\times 3}$$

Divide \frac{5-\sqrt{649}}{3} by 2.

$$x=\frac{5-\sqrt{649}}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{649}+5}{6} x=\frac{5-\sqrt{649}}{6}$$

Steps for Completing the Square

Use the distributive property to multiply x-1 by x-\frac{2}{3} and combine like terms.

$$x^{2}-\frac{5}{3}x+\frac{2}{3}=18$$

Subtract \frac{2}{3} from both sides.

$$x^{2}-\frac{5}{3}x=18-\frac{2}{3}$$

Subtract \frac{2}{3} from 18 to get \frac{52}{3}.

$$x^{2}-\frac{5}{3}x=\frac{52}{3}$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=\frac{52}{3}+\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{52}{3}+\frac{25}{36}$$

Add \frac{52}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{649}{36}$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{649}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{649}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{\sqrt{649}}{6} x-\frac{5}{6}=-\frac{\sqrt{649}}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$x=\frac{\sqrt{649}+5}{6} x=\frac{5-\sqrt{649}}{6}$$