Solve 3y(y-1)=2(y-1)

$$3 y ( y - 1 ) = 2 ( y - 1 )$$

Solve for y

$$y=1$$

$$y=\frac{2}{3}\approx 0.666666667$$

Steps Using the Quadratic Formula

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

$$3y^{2}-3y=2\left(y-1\right)$$

Use the distributive property to multiply 2 by y-1.

$$3y^{2}-3y=2y-2$$

Subtract 2y from both sides.

$$3y^{2}-3y-2y=-2$$

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

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

Add 2 to both sides.

$$3y^{2}-5y+2=0$$

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

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

Square -5.

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

Multiply -4 times 3.

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

Multiply -12 times 2.

$$y=\frac{-\left(-5\right)±\sqrt{25-24}}{2\times 3}$$

Add 25 to -24.

$$y=\frac{-\left(-5\right)±\sqrt{1}}{2\times 3}$$

Take the square root of 1.

$$y=\frac{-\left(-5\right)±1}{2\times 3}$$

The opposite of -5 is 5.

$$y=\frac{5±1}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{5±1}{6}$$

Now solve the equation y=\frac{5±1}{6} when ± is plus. Add 5 to 1.

$$y=\frac{6}{6}$$

Divide 6 by 6.

$$y=1$$

Now solve the equation y=\frac{5±1}{6} when ± is minus. Subtract 1 from 5.

$$y=\frac{4}{6}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$y=\frac{2}{3}$$

The equation is now solved.

$$y=1 y=\frac{2}{3}$$

Steps for Completing the Square

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

$$3y^{2}-3y=2\left(y-1\right)$$

Use the distributive property to multiply 2 by y-1.

$$3y^{2}-3y=2y-2$$

Subtract 2y from both sides.

$$3y^{2}-3y-2y=-2$$

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

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

Divide both sides by 3.

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

Dividing by 3 undoes the multiplication by 3.

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

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.

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

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

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

Add -\frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{5}{3}y+\frac{25}{36}=\frac{1}{36}$$

Factor y^{2}-\frac{5}{3}y+\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}.

$$\left(y-\frac{5}{6}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{5}{6}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$y-\frac{5}{6}=\frac{1}{6} y-\frac{5}{6}=-\frac{1}{6}$$

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

$$y=1 y=\frac{2}{3}$$