Solve 3y(4y-3)=8y-6

$$3 y ( 4 y - 3 ) = 8 y - 6$$

Solve for y

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

$$y=\frac{3}{4}=0.75$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3y by 4y-3.

$$12y^{2}-9y=8y-6$$

Subtract 8y from both sides.

$$12y^{2}-9y-8y=-6$$

Combine -9y and -8y to get -17y.

$$12y^{2}-17y=-6$$

Add 6 to both sides.

$$12y^{2}-17y+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -17 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 12\times 6}}{2\times 12}$$

Square -17.

$$y=\frac{-\left(-17\right)±\sqrt{289-4\times 12\times 6}}{2\times 12}$$

Multiply -4 times 12.

$$y=\frac{-\left(-17\right)±\sqrt{289-48\times 6}}{2\times 12}$$

Multiply -48 times 6.

$$y=\frac{-\left(-17\right)±\sqrt{289-288}}{2\times 12}$$

Add 289 to -288.

$$y=\frac{-\left(-17\right)±\sqrt{1}}{2\times 12}$$

Take the square root of 1.

$$y=\frac{-\left(-17\right)±1}{2\times 12}$$

The opposite of -17 is 17.

$$y=\frac{17±1}{2\times 12}$$

Multiply 2 times 12.

$$y=\frac{17±1}{24}$$

Now solve the equation y=\frac{17±1}{24} when ± is plus. Add 17 to 1.

$$y=\frac{18}{24}$$

Reduce the fraction \frac{18}{24} to lowest terms by extracting and canceling out 6.

$$y=\frac{3}{4}$$

Now solve the equation y=\frac{17±1}{24} when ± is minus. Subtract 1 from 17.

$$y=\frac{16}{24}$$

Reduce the fraction \frac{16}{24} to lowest terms by extracting and canceling out 8.

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

The equation is now solved.

$$y=\frac{3}{4} y=\frac{2}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 3y by 4y-3.

$$12y^{2}-9y=8y-6$$

Subtract 8y from both sides.

$$12y^{2}-9y-8y=-6$$

Combine -9y and -8y to get -17y.

$$12y^{2}-17y=-6$$

Divide both sides by 12.

$$\frac{12y^{2}-17y}{12}=-\frac{6}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$y^{2}-\frac{17}{12}y=-\frac{6}{12}$$

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

$$y^{2}-\frac{17}{12}y=-\frac{1}{2}$$

Divide -\frac{17}{12}, the coefficient of the x term, by 2 to get -\frac{17}{24}. Then add the square of -\frac{17}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{17}{12}y+\left(-\frac{17}{24}\right)^{2}=-\frac{1}{2}+\left(-\frac{17}{24}\right)^{2}$$

Square -\frac{17}{24} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{17}{12}y+\frac{289}{576}=-\frac{1}{2}+\frac{289}{576}$$

Add -\frac{1}{2} to \frac{289}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{17}{12}y+\frac{289}{576}=\frac{1}{576}$$

Factor y^{2}-\frac{17}{12}y+\frac{289}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{17}{24}\right)^{2}=\frac{1}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{17}{24}\right)^{2}}=\sqrt{\frac{1}{576}}$$

Simplify.

$$y-\frac{17}{24}=\frac{1}{24} y-\frac{17}{24}=-\frac{1}{24}$$

Add \frac{17}{24} to both sides of the equation.

$$y=\frac{3}{4} y=\frac{2}{3}$$