Solve 34-y=120/y

$$34 - y = \frac { 120 } { y }$$

Solve for y

$$y=4$$

$$y=30$$

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.

$$y\times 34-yy=120$$

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

$$y\times 34-y^{2}=120$$

Subtract 120 from both sides.

$$y\times 34-y^{2}-120=0$$

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.

$$-y^{2}+34y-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 34 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-34±\sqrt{34^{2}-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

Square 34.

$$y=\frac{-34±\sqrt{1156-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-34±\sqrt{1156+4\left(-120\right)}}{2\left(-1\right)}$$

Multiply 4 times -120.

$$y=\frac{-34±\sqrt{1156-480}}{2\left(-1\right)}$$

Add 1156 to -480.

$$y=\frac{-34±\sqrt{676}}{2\left(-1\right)}$$

Take the square root of 676.

$$y=\frac{-34±26}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-34±26}{-2}$$

Now solve the equation y=\frac{-34±26}{-2} when ± is plus. Add -34 to 26.

$$y=-\frac{8}{-2}$$

Divide -8 by -2.

$$y=4$$

Now solve the equation y=\frac{-34±26}{-2} when ± is minus. Subtract 26 from -34.

$$y=-\frac{60}{-2}$$

Divide -60 by -2.

$$y=30$$

The equation is now solved.

$$y=4 y=30$$

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.

$$y\times 34-yy=120$$

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

$$y\times 34-y^{2}=120$$

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.

$$-y^{2}+34y=120$$

Divide both sides by -1.

$$\frac{-y^{2}+34y}{-1}=\frac{120}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{34}{-1}y=\frac{120}{-1}$$

Divide 34 by -1.

$$y^{2}-34y=\frac{120}{-1}$$

Divide 120 by -1.

$$y^{2}-34y=-120$$

Divide -34, the coefficient of the x term, by 2 to get -17. Then add the square of -17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-34y+\left(-17\right)^{2}=-120+\left(-17\right)^{2}$$

Square -17.

$$y^{2}-34y+289=-120+289$$

Add -120 to 289.

$$y^{2}-34y+289=169$$

Factor y^{2}-34y+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-17\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-17\right)^{2}}=\sqrt{169}$$

Simplify.

$$y-17=13 y-17=-13$$

Add 17 to both sides of the equation.

$$y=30 y=4$$