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

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

Solve for y

$$y=-1$$

$$y=\frac{1}{6}\approx 0.166666667$$

Steps Using the Quadratic Formula

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

$$6y^{2}-2y+7y=1$$

Combine -2y and 7y to get 5y.

$$6y^{2}+5y=1$$

Subtract 1 from both sides.

$$6y^{2}+5y-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-5±\sqrt{5^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square 5.

$$y=\frac{-5±\sqrt{25-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-5±\sqrt{25-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$y=\frac{-5±\sqrt{25+24}}{2\times 6}$$

Add 25 to 24.

$$y=\frac{-5±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$y=\frac{-5±7}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{-5±7}{12}$$

Now solve the equation y=\frac{-5±7}{12} when ± is plus. Add -5 to 7.

$$y=\frac{2}{12}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

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

Now solve the equation y=\frac{-5±7}{12} when ± is minus. Subtract 7 from -5.

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

Divide -12 by 12.

$$y=-1$$

The equation is now solved.

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

Steps for Completing the Square

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

$$6y^{2}-2y+7y=1$$

Combine -2y and 7y to get 5y.

$$6y^{2}+5y=1$$

Divide both sides by 6.

$$\frac{6y^{2}+5y}{6}=\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$y^{2}+\frac{5}{6}y=\frac{1}{6}$$

Divide \frac{5}{6}, the coefficient of the x term, by 2 to get \frac{5}{12}. Then add the square of \frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{5}{6}y+\left(\frac{5}{12}\right)^{2}=\frac{1}{6}+\left(\frac{5}{12}\right)^{2}$$

Square \frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{5}{6}y+\frac{25}{144}=\frac{1}{6}+\frac{25}{144}$$

Add \frac{1}{6} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{5}{6}y+\frac{25}{144}=\frac{49}{144}$$

Factor y^{2}+\frac{5}{6}y+\frac{25}{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}.

$$\left(y+\frac{5}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{5}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$y+\frac{5}{12}=\frac{7}{12} y+\frac{5}{12}=-\frac{7}{12}$$

Subtract \frac{5}{12} from both sides of the equation.

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