Solve 3x(x-1)=2x+5

$$3 x ( x - 1 ) = 2 x + 5$$

Solve for x

$$x = \frac{\sqrt{85} + 5}{6} \approx 2.369924076$$

$$x=\frac{5-\sqrt{85}}{6}\approx -0.70325741$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x-1.

$$3x^{2}-3x=2x+5$$

Subtract 2x from both sides.

$$3x^{2}-3x-2x=5$$

Combine -3x and -2x to get -5x.

$$3x^{2}-5x=5$$

Subtract 5 from both sides.

$$3x^{2}-5x-5=0$$

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

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

Square -5.

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

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-5\right)}}{2\times 3}$$

Multiply -12 times -5.

$$x=\frac{-\left(-5\right)±\sqrt{25+60}}{2\times 3}$$

Add 25 to 60.

$$x=\frac{-\left(-5\right)±\sqrt{85}}{2\times 3}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{85}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±\sqrt{85}}{6}$$

Now solve the equation x=\frac{5±\sqrt{85}}{6} when ± is plus. Add 5 to \sqrt{85}.

$$x=\frac{\sqrt{85}+5}{6}$$

Now solve the equation x=\frac{5±\sqrt{85}}{6} when ± is minus. Subtract \sqrt{85} from 5.

$$x=\frac{5-\sqrt{85}}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{85}+5}{6} x=\frac{5-\sqrt{85}}{6}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x-1.

$$3x^{2}-3x=2x+5$$

Subtract 2x from both sides.

$$3x^{2}-3x-2x=5$$

Combine -3x and -2x to get -5x.

$$3x^{2}-5x=5$$

Divide both sides by 3.

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

Dividing by 3 undoes the multiplication by 3.

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

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.

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

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

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

Add \frac{5}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{85}{36}$$

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}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{85}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{85}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{\sqrt{85}}{6} x-\frac{5}{6}=-\frac{\sqrt{85}}{6}$$

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

$$x=\frac{\sqrt{85}+5}{6} x=\frac{5-\sqrt{85}}{6}$$