Solve 1/2y-5/6=1/4y+2

$$\frac{ 1 }{ 2 } y- \frac{ 5 }{ 6 } = \frac{ 1 }{ 4y } +2$$

Solve for y

$$y = \frac{\sqrt{307} + 17}{6} \approx 5.753569245$$

$$y=\frac{17-\sqrt{307}}{6}\approx -0.086902578$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12y, the least common multiple of 2,6,4y.

$$\frac{1}{2}y\times 12y+12y\left(-\frac{5}{6}\right)=3+12y\times 2$$

Multiply \frac{1}{2} and 12 to get 6.

$$6yy+12y\left(-\frac{5}{6}\right)=3+12y\times 2$$

Multiply y and y to get y^{2}.

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

Multiply 12 and -\frac{5}{6} to get -10.

$$6y^{2}-10y=3+12y\times 2$$

Multiply 12 and 2 to get 24.

$$6y^{2}-10y=3+24y$$

Subtract 3 from both sides.

$$6y^{2}-10y-3=24y$$

Subtract 24y from both sides.

$$6y^{2}-10y-3-24y=0$$

Combine -10y and -24y to get -34y.

$$6y^{2}-34y-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -34 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 6\left(-3\right)}}{2\times 6}$$

Square -34.

$$y=\frac{-\left(-34\right)±\sqrt{1156-4\times 6\left(-3\right)}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-\left(-34\right)±\sqrt{1156-24\left(-3\right)}}{2\times 6}$$

Multiply -24 times -3.

$$y=\frac{-\left(-34\right)±\sqrt{1156+72}}{2\times 6}$$

Add 1156 to 72.

$$y=\frac{-\left(-34\right)±\sqrt{1228}}{2\times 6}$$

Take the square root of 1228.

$$y=\frac{-\left(-34\right)±2\sqrt{307}}{2\times 6}$$

The opposite of -34 is 34.

$$y=\frac{34±2\sqrt{307}}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{34±2\sqrt{307}}{12}$$

Now solve the equation y=\frac{34±2\sqrt{307}}{12} when ± is plus. Add 34 to 2\sqrt{307}.

$$y=\frac{2\sqrt{307}+34}{12}$$

Divide 34+2\sqrt{307} by 12.

$$y=\frac{\sqrt{307}+17}{6}$$

Now solve the equation y=\frac{34±2\sqrt{307}}{12} when ± is minus. Subtract 2\sqrt{307} from 34.

$$y=\frac{34-2\sqrt{307}}{12}$$

Divide 34-2\sqrt{307} by 12.

$$y=\frac{17-\sqrt{307}}{6}$$

The equation is now solved.

$$y=\frac{\sqrt{307}+17}{6} y=\frac{17-\sqrt{307}}{6}$$

Steps for Completing the Square

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12y, the least common multiple of 2,6,4y.

$$\frac{1}{2}y\times 12y+12y\left(-\frac{5}{6}\right)=3+12y\times 2$$

Multiply \frac{1}{2} and 12 to get 6.

$$6yy+12y\left(-\frac{5}{6}\right)=3+12y\times 2$$

Multiply y and y to get y^{2}.

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

Multiply 12 and -\frac{5}{6} to get -10.

$$6y^{2}-10y=3+12y\times 2$$

Multiply 12 and 2 to get 24.

$$6y^{2}-10y=3+24y$$

Subtract 24y from both sides.

$$6y^{2}-10y-24y=3$$

Combine -10y and -24y to get -34y.

$$6y^{2}-34y=3$$

Divide both sides by 6.

$$\frac{6y^{2}-34y}{6}=\frac{3}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$y^{2}+\left(-\frac{34}{6}\right)y=\frac{3}{6}$$

Reduce the fraction \frac{-34}{6} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{17}{3}y=\frac{3}{6}$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$y^{2}-\frac{17}{3}y=\frac{1}{2}$$

Divide -\frac{17}{3}, the coefficient of the x term, by 2 to get -\frac{17}{6}. Then add the square of -\frac{17}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{17}{3}y+\left(-\frac{17}{6}\right)^{2}=\frac{1}{2}+\left(-\frac{17}{6}\right)^{2}$$

Square -\frac{17}{6} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{17}{3}y+\frac{289}{36}=\frac{1}{2}+\frac{289}{36}$$

Add \frac{1}{2} to \frac{289}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{17}{3}y+\frac{289}{36}=\frac{307}{36}$$

Factor y^{2}-\frac{17}{3}y+\frac{289}{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}.

$$\left(y-\frac{17}{6}\right)^{2}=\frac{307}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{17}{6}\right)^{2}}=\sqrt{\frac{307}{36}}$$

Simplify.

$$y-\frac{17}{6}=\frac{\sqrt{307}}{6} y-\frac{17}{6}=-\frac{\sqrt{307}}{6}$$

Add \frac{17}{6} to both sides of the equation.

$$y=\frac{\sqrt{307}+17}{6} y=\frac{17-\sqrt{307}}{6}$$