Solve 3y(2y-1)=4y-2

$$3 y ( 2 y - 1 ) = 4 y - 2$$

Solve for y

$$y=\frac{1}{2}=0.5$$

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

Steps Using the Quadratic Formula

Use the distributive property to multiply 3y by 2y-1.

$$6y^{2}-3y=4y-2$$

Subtract 4y from both sides.

$$6y^{2}-3y-4y=-2$$

Combine -3y and -4y to get -7y.

$$6y^{2}-7y=-2$$

Add 2 to both sides.

$$6y^{2}-7y+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -7 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\times 2}}{2\times 6}$$

Square -7.

$$y=\frac{-\left(-7\right)±\sqrt{49-4\times 6\times 2}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-\left(-7\right)±\sqrt{49-24\times 2}}{2\times 6}$$

Multiply -24 times 2.

$$y=\frac{-\left(-7\right)±\sqrt{49-48}}{2\times 6}$$

Add 49 to -48.

$$y=\frac{-\left(-7\right)±\sqrt{1}}{2\times 6}$$

Take the square root of 1.

$$y=\frac{-\left(-7\right)±1}{2\times 6}$$

The opposite of -7 is 7.

$$y=\frac{7±1}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{7±1}{12}$$

Now solve the equation y=\frac{7±1}{12} when ± is plus. Add 7 to 1.

$$y=\frac{8}{12}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

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

Now solve the equation y=\frac{7±1}{12} when ± is minus. Subtract 1 from 7.

$$y=\frac{6}{12}$$

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 3y by 2y-1.

$$6y^{2}-3y=4y-2$$

Subtract 4y from both sides.

$$6y^{2}-3y-4y=-2$$

Combine -3y and -4y to get -7y.

$$6y^{2}-7y=-2$$

Divide both sides by 6.

$$\frac{6y^{2}-7y}{6}=-\frac{2}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$y^{2}-\frac{7}{6}y=-\frac{2}{6}$$

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

$$y^{2}-\frac{7}{6}y=-\frac{1}{3}$$

Divide -\frac{7}{6}, the coefficient of the x term, by 2 to get -\frac{7}{12}. Then add the square of -\frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{7}{6}y+\left(-\frac{7}{12}\right)^{2}=-\frac{1}{3}+\left(-\frac{7}{12}\right)^{2}$$

Square -\frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{7}{6}y+\frac{49}{144}=-\frac{1}{3}+\frac{49}{144}$$

Add -\frac{1}{3} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{7}{6}y+\frac{49}{144}=\frac{1}{144}$$

Factor y^{2}-\frac{7}{6}y+\frac{49}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{7}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{7}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$y-\frac{7}{12}=\frac{1}{12} y-\frac{7}{12}=-\frac{1}{12}$$

Add \frac{7}{12} to both sides of the equation.

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