Solve 30-6y+y^2/30y-6y^2=0

$$\frac{ 30-6y+ { y }^{ 2 } }{ 30y-6 { y }^{ 2 } } = 0$$

Solve for y

$$y=3+\sqrt{21}i\approx 3+4.582575695i$$

$$y=-\sqrt{21}i+3\approx 3-4.582575695i$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 6y\left(-y+5\right).

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 30}}{2}$$

Square -6.

$$y=\frac{-\left(-6\right)±\sqrt{36-4\times 30}}{2}$$

Multiply -4 times 30.

$$y=\frac{-\left(-6\right)±\sqrt{36-120}}{2}$$

Add 36 to -120.

$$y=\frac{-\left(-6\right)±\sqrt{-84}}{2}$$

Take the square root of -84.

$$y=\frac{-\left(-6\right)±2\sqrt{21}i}{2}$$

The opposite of -6 is 6.

$$y=\frac{6±2\sqrt{21}i}{2}$$

Now solve the equation y=\frac{6±2\sqrt{21}i}{2} when ± is plus. Add 6 to 2i\sqrt{21}.

$$y=\frac{6+2\sqrt{21}i}{2}$$

Divide 6+2i\sqrt{21} by 2.

$$y=3+\sqrt{21}i$$

Now solve the equation y=\frac{6±2\sqrt{21}i}{2} when ± is minus. Subtract 2i\sqrt{21} from 6.

$$y=\frac{-2\sqrt{21}i+6}{2}$$

Divide 6-2i\sqrt{21} by 2.

$$y=-\sqrt{21}i+3$$

The equation is now solved.

$$y=3+\sqrt{21}i y=-\sqrt{21}i+3$$

Steps for Completing the Square

Variable y cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 6y\left(-y+5\right).

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

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

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

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.

$$y^{2}-6y=-30$$

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

$$y^{2}-6y+\left(-3\right)^{2}=-30+\left(-3\right)^{2}$$

Square -3.

$$y^{2}-6y+9=-30+9$$

Add -30 to 9.

$$y^{2}-6y+9=-21$$

Factor y^{2}-6y+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-3\right)^{2}=-21$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-3\right)^{2}}=\sqrt{-21}$$

Simplify.

$$y-3=\sqrt{21}i y-3=-\sqrt{21}i$$

Add 3 to both sides of the equation.

$$y=3+\sqrt{21}i y=-\sqrt{21}i+3$$