Solve 2/2-3y+1/y=5/6

$$\frac { 2 } { 2 - 3 y } + \frac { 1 } { y } = \frac { 5 } { 6 }$$

Solve for y

$$y=\frac{8+2\sqrt{29}i}{15}\approx 0.533333333+0.718021974i$$

$$y=\frac{-2\sqrt{29}i+8}{15}\approx 0.533333333-0.718021974i$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values 0,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 6y\left(3y-2\right), the least common multiple of 2-3y,y,6.

$$-6y\times 2+18y-12=5y\left(3y-2\right)$$

Multiply -6 and 2 to get -12.

$$-12y+18y-12=5y\left(3y-2\right)$$

Combine -12y and 18y to get 6y.

$$6y-12=5y\left(3y-2\right)$$

Use the distributive property to multiply 5y by 3y-2.

$$6y-12=15y^{2}-10y$$

Subtract 15y^{2} from both sides.

$$6y-12-15y^{2}=-10y$$

Add 10y to both sides.

$$6y-12-15y^{2}+10y=0$$

Combine 6y and 10y to get 16y.

$$16y-12-15y^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-15y^{2}+16y-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 16 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-16±\sqrt{16^{2}-4\left(-15\right)\left(-12\right)}}{2\left(-15\right)}$$

Square 16.

$$y=\frac{-16±\sqrt{256-4\left(-15\right)\left(-12\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$y=\frac{-16±\sqrt{256+60\left(-12\right)}}{2\left(-15\right)}$$

Multiply 60 times -12.

$$y=\frac{-16±\sqrt{256-720}}{2\left(-15\right)}$$

Add 256 to -720.

$$y=\frac{-16±\sqrt{-464}}{2\left(-15\right)}$$

Take the square root of -464.

$$y=\frac{-16±4\sqrt{29}i}{2\left(-15\right)}$$

Multiply 2 times -15.

$$y=\frac{-16±4\sqrt{29}i}{-30}$$

Now solve the equation y=\frac{-16±4\sqrt{29}i}{-30} when ± is plus. Add -16 to 4i\sqrt{29}.

$$y=\frac{-16+4\sqrt{29}i}{-30}$$

Divide -16+4i\sqrt{29} by -30.

$$y=\frac{-2\sqrt{29}i+8}{15}$$

Now solve the equation y=\frac{-16±4\sqrt{29}i}{-30} when ± is minus. Subtract 4i\sqrt{29} from -16.

$$y=\frac{-4\sqrt{29}i-16}{-30}$$

Divide -16-4i\sqrt{29} by -30.

$$y=\frac{8+2\sqrt{29}i}{15}$$

The equation is now solved.

$$y=\frac{-2\sqrt{29}i+8}{15} y=\frac{8+2\sqrt{29}i}{15}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values 0,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 6y\left(3y-2\right), the least common multiple of 2-3y,y,6.

$$-6y\times 2+18y-12=5y\left(3y-2\right)$$

Multiply -6 and 2 to get -12.

$$-12y+18y-12=5y\left(3y-2\right)$$

Combine -12y and 18y to get 6y.

$$6y-12=5y\left(3y-2\right)$$

Use the distributive property to multiply 5y by 3y-2.

$$6y-12=15y^{2}-10y$$

Subtract 15y^{2} from both sides.

$$6y-12-15y^{2}=-10y$$

Add 10y to both sides.

$$6y-12-15y^{2}+10y=0$$

Combine 6y and 10y to get 16y.

$$16y-12-15y^{2}=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$16y-15y^{2}=12$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-15y^{2}+16y=12$$

Divide both sides by -15.

$$\frac{-15y^{2}+16y}{-15}=\frac{12}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$y^{2}+\frac{16}{-15}y=\frac{12}{-15}$$

Divide 16 by -15.

$$y^{2}-\frac{16}{15}y=\frac{12}{-15}$$

Reduce the fraction \frac{12}{-15} to lowest terms by extracting and canceling out 3.

$$y^{2}-\frac{16}{15}y=-\frac{4}{5}$$

Divide -\frac{16}{15}, the coefficient of the x term, by 2 to get -\frac{8}{15}. Then add the square of -\frac{8}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{16}{15}y+\left(-\frac{8}{15}\right)^{2}=-\frac{4}{5}+\left(-\frac{8}{15}\right)^{2}$$

Square -\frac{8}{15} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{16}{15}y+\frac{64}{225}=-\frac{4}{5}+\frac{64}{225}$$

Add -\frac{4}{5} to \frac{64}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{16}{15}y+\frac{64}{225}=-\frac{116}{225}$$

Factor y^{2}-\frac{16}{15}y+\frac{64}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{8}{15}\right)^{2}=-\frac{116}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{8}{15}\right)^{2}}=\sqrt{-\frac{116}{225}}$$

Simplify.

$$y-\frac{8}{15}=\frac{2\sqrt{29}i}{15} y-\frac{8}{15}=-\frac{2\sqrt{29}i}{15}$$

Add \frac{8}{15} to both sides of the equation.

$$y=\frac{8+2\sqrt{29}i}{15} y=\frac{-2\sqrt{29}i+8}{15}$$