Solve y+2/y=135

$$y+ \frac{ 2 }{ y } =135$$

Solve for y

$$y = \frac{\sqrt{18217} + 135}{2} \approx 134.985183559$$

$$y=\frac{135-\sqrt{18217}}{2}\approx 0.014816441$$

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 y.

$$yy+2=135y$$

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

$$y^{2}+2=135y$$

Subtract 135y from both sides.

$$y^{2}+2-135y=0$$

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.

$$y^{2}-135y+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -135 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-135\right)±\sqrt{\left(-135\right)^{2}-4\times 2}}{2}$$

Square -135.

$$y=\frac{-\left(-135\right)±\sqrt{18225-4\times 2}}{2}$$

Multiply -4 times 2.

$$y=\frac{-\left(-135\right)±\sqrt{18225-8}}{2}$$

Add 18225 to -8.

$$y=\frac{-\left(-135\right)±\sqrt{18217}}{2}$$

The opposite of -135 is 135.

$$y=\frac{135±\sqrt{18217}}{2}$$

Now solve the equation y=\frac{135±\sqrt{18217}}{2} when ± is plus. Add 135 to \sqrt{18217}.

$$y=\frac{\sqrt{18217}+135}{2}$$

Now solve the equation y=\frac{135±\sqrt{18217}}{2} when ± is minus. Subtract \sqrt{18217} from 135.

$$y=\frac{135-\sqrt{18217}}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{18217}+135}{2} y=\frac{135-\sqrt{18217}}{2}$$

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 y.

$$yy+2=135y$$

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

$$y^{2}+2=135y$$

Subtract 135y from both sides.

$$y^{2}+2-135y=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}-135y=-2$$

Divide -135, the coefficient of the x term, by 2 to get -\frac{135}{2}. Then add the square of -\frac{135}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-135y+\left(-\frac{135}{2}\right)^{2}=-2+\left(-\frac{135}{2}\right)^{2}$$

Square -\frac{135}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-135y+\frac{18225}{4}=-2+\frac{18225}{4}$$

Add -2 to \frac{18225}{4}.

$$y^{2}-135y+\frac{18225}{4}=\frac{18217}{4}$$

Factor y^{2}-135y+\frac{18225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{135}{2}\right)^{2}=\frac{18217}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{135}{2}\right)^{2}}=\sqrt{\frac{18217}{4}}$$

Simplify.

$$y-\frac{135}{2}=\frac{\sqrt{18217}}{2} y-\frac{135}{2}=-\frac{\sqrt{18217}}{2}$$

Add \frac{135}{2} to both sides of the equation.

$$y=\frac{\sqrt{18217}+135}{2} y=\frac{135-\sqrt{18217}}{2}$$