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

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

Solve for y

$$y=\frac{\sqrt{159}}{3}+4\approx 8.203173404$$

$$y=-\frac{\sqrt{159}}{3}+4\approx -0.203173404$$

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 5y, the least common multiple of y,5.

$$5\left(y-1\right)+y\left(3y+1\right)=30y$$

Use the distributive property to multiply 5 by y-1.

$$5y-5+y\left(3y+1\right)=30y$$

Use the distributive property to multiply y by 3y+1.

$$5y-5+3y^{2}+y=30y$$

Combine 5y and y to get 6y.

$$6y-5+3y^{2}=30y$$

Subtract 30y from both sides.

$$6y-5+3y^{2}-30y=0$$

Combine 6y and -30y to get -24y.

$$-24y-5+3y^{2}=0$$

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.

$$3y^{2}-24y-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -24 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

Square -24.

$$y=\frac{-\left(-24\right)±\sqrt{576-4\times 3\left(-5\right)}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-\left(-24\right)±\sqrt{576-12\left(-5\right)}}{2\times 3}$$

Multiply -12 times -5.

$$y=\frac{-\left(-24\right)±\sqrt{576+60}}{2\times 3}$$

Add 576 to 60.

$$y=\frac{-\left(-24\right)±\sqrt{636}}{2\times 3}$$

Take the square root of 636.

$$y=\frac{-\left(-24\right)±2\sqrt{159}}{2\times 3}$$

The opposite of -24 is 24.

$$y=\frac{24±2\sqrt{159}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{24±2\sqrt{159}}{6}$$

Now solve the equation y=\frac{24±2\sqrt{159}}{6} when ± is plus. Add 24 to 2\sqrt{159}.

$$y=\frac{2\sqrt{159}+24}{6}$$

Divide 24+2\sqrt{159} by 6.

$$y=\frac{\sqrt{159}}{3}+4$$

Now solve the equation y=\frac{24±2\sqrt{159}}{6} when ± is minus. Subtract 2\sqrt{159} from 24.

$$y=\frac{24-2\sqrt{159}}{6}$$

Divide 24-2\sqrt{159} by 6.

$$y=-\frac{\sqrt{159}}{3}+4$$

The equation is now solved.

$$y=\frac{\sqrt{159}}{3}+4 y=-\frac{\sqrt{159}}{3}+4$$

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 5y, the least common multiple of y,5.

$$5\left(y-1\right)+y\left(3y+1\right)=30y$$

Use the distributive property to multiply 5 by y-1.

$$5y-5+y\left(3y+1\right)=30y$$

Use the distributive property to multiply y by 3y+1.

$$5y-5+3y^{2}+y=30y$$

Combine 5y and y to get 6y.

$$6y-5+3y^{2}=30y$$

Subtract 30y from both sides.

$$6y-5+3y^{2}-30y=0$$

Combine 6y and -30y to get -24y.

$$-24y-5+3y^{2}=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$-24y+3y^{2}=5$$

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.

$$3y^{2}-24y=5$$

Divide both sides by 3.

$$\frac{3y^{2}-24y}{3}=\frac{5}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$y^{2}+\left(-\frac{24}{3}\right)y=\frac{5}{3}$$

Divide -24 by 3.

$$y^{2}-8y=\frac{5}{3}$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-8y+\left(-4\right)^{2}=\frac{5}{3}+\left(-4\right)^{2}$$

Square -4.

$$y^{2}-8y+16=\frac{5}{3}+16$$

Add \frac{5}{3} to 16.

$$y^{2}-8y+16=\frac{53}{3}$$

Factor y^{2}-8y+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-4\right)^{2}=\frac{53}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-4\right)^{2}}=\sqrt{\frac{53}{3}}$$

Simplify.

$$y-4=\frac{\sqrt{159}}{3} y-4=-\frac{\sqrt{159}}{3}$$

Add 4 to both sides of the equation.

$$y=\frac{\sqrt{159}}{3}+4 y=-\frac{\sqrt{159}}{3}+4$$