Solve (2y-1)(3y+7)=3

$$( 2 y - 1 ) ( 3 y + 7 ) = 3$$

Solve for y

$$y = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

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

Steps Using the Quadratic Formula

Use the distributive property to multiply 2y-1 by 3y+7 and combine like terms.

$$6y^{2}+11y-7=3$$

Subtract 3 from both sides.

$$6y^{2}+11y-7-3=0$$

Subtract 3 from -7 to get -10.

$$6y^{2}+11y-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 11 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-11±\sqrt{11^{2}-4\times 6\left(-10\right)}}{2\times 6}$$

Square 11.

$$y=\frac{-11±\sqrt{121-4\times 6\left(-10\right)}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-11±\sqrt{121-24\left(-10\right)}}{2\times 6}$$

Multiply -24 times -10.

$$y=\frac{-11±\sqrt{121+240}}{2\times 6}$$

Add 121 to 240.

$$y=\frac{-11±\sqrt{361}}{2\times 6}$$

Take the square root of 361.

$$y=\frac{-11±19}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{-11±19}{12}$$

Now solve the equation y=\frac{-11±19}{12} when ± is plus. Add -11 to 19.

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

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{-11±19}{12} when ± is minus. Subtract 19 from -11.

$$y=-\frac{30}{12}$$

Reduce the fraction \frac{-30}{12} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 2y-1 by 3y+7 and combine like terms.

$$6y^{2}+11y-7=3$$

Add 7 to both sides.

$$6y^{2}+11y=3+7$$

Add 3 and 7 to get 10.

$$6y^{2}+11y=10$$

Divide both sides by 6.

$$\frac{6y^{2}+11y}{6}=\frac{10}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$y^{2}+\frac{11}{6}y=\frac{10}{6}$$

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{11}{6}y=\frac{5}{3}$$

Divide \frac{11}{6}, the coefficient of the x term, by 2 to get \frac{11}{12}. Then add the square of \frac{11}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{11}{6}y+\left(\frac{11}{12}\right)^{2}=\frac{5}{3}+\left(\frac{11}{12}\right)^{2}$$

Square \frac{11}{12} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{11}{6}y+\frac{121}{144}=\frac{5}{3}+\frac{121}{144}$$

Add \frac{5}{3} to \frac{121}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{11}{6}y+\frac{121}{144}=\frac{361}{144}$$

Factor y^{2}+\frac{11}{6}y+\frac{121}{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}.

$$\left(y+\frac{11}{12}\right)^{2}=\frac{361}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{11}{12}\right)^{2}}=\sqrt{\frac{361}{144}}$$

Simplify.

$$y+\frac{11}{12}=\frac{19}{12} y+\frac{11}{12}=-\frac{19}{12}$$

Subtract \frac{11}{12} from both sides of the equation.

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