Solve 2(y+3)-6(y+6)=2+y(y-6)

$$2 ( y + 3 ) - 6 ( y + 6 ) = 2 + y ( y - 6 )$$

Solve for y

$$y=-\sqrt{31}i+1\approx 1-5.567764363i$$

$$y=1+\sqrt{31}i\approx 1+5.567764363i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by y+3.

$$2y+6-6\left(y+6\right)=2+y\left(y-6\right)$$

Use the distributive property to multiply -6 by y+6.

$$2y+6-6y-36=2+y\left(y-6\right)$$

Combine 2y and -6y to get -4y.

$$-4y+6-36=2+y\left(y-6\right)$$

Subtract 36 from 6 to get -30.

$$-4y-30=2+y\left(y-6\right)$$

Use the distributive property to multiply y by y-6.

$$-4y-30=2+y^{2}-6y$$

Subtract 2 from both sides.

$$-4y-30-2=y^{2}-6y$$

Subtract 2 from -30 to get -32.

$$-4y-32=y^{2}-6y$$

Subtract y^{2} from both sides.

$$-4y-32-y^{2}=-6y$$

Add 6y to both sides.

$$-4y-32-y^{2}+6y=0$$

Combine -4y and 6y to get 2y.

$$2y-32-y^{2}=0$$

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.

$$-y^{2}+2y-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-32\right)}}{2\left(-1\right)}$$

Square 2.

$$y=\frac{-2±\sqrt{4-4\left(-1\right)\left(-32\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-2±\sqrt{4+4\left(-32\right)}}{2\left(-1\right)}$$

Multiply 4 times -32.

$$y=\frac{-2±\sqrt{4-128}}{2\left(-1\right)}$$

Add 4 to -128.

$$y=\frac{-2±\sqrt{-124}}{2\left(-1\right)}$$

Take the square root of -124.

$$y=\frac{-2±2\sqrt{31}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-2±2\sqrt{31}i}{-2}$$

Now solve the equation y=\frac{-2±2\sqrt{31}i}{-2} when ± is plus. Add -2 to 2i\sqrt{31}.

$$y=\frac{-2+2\sqrt{31}i}{-2}$$

Divide -2+2i\sqrt{31} by -2.

$$y=-\sqrt{31}i+1$$

Now solve the equation y=\frac{-2±2\sqrt{31}i}{-2} when ± is minus. Subtract 2i\sqrt{31} from -2.

$$y=\frac{-2\sqrt{31}i-2}{-2}$$

Divide -2-2i\sqrt{31} by -2.

$$y=1+\sqrt{31}i$$

The equation is now solved.

$$y=-\sqrt{31}i+1 y=1+\sqrt{31}i$$

Steps for Completing the Square

Use the distributive property to multiply 2 by y+3.

$$2y+6-6\left(y+6\right)=2+y\left(y-6\right)$$

Use the distributive property to multiply -6 by y+6.

$$2y+6-6y-36=2+y\left(y-6\right)$$

Combine 2y and -6y to get -4y.

$$-4y+6-36=2+y\left(y-6\right)$$

Subtract 36 from 6 to get -30.

$$-4y-30=2+y\left(y-6\right)$$

Use the distributive property to multiply y by y-6.

$$-4y-30=2+y^{2}-6y$$

Subtract y^{2} from both sides.

$$-4y-30-y^{2}=2-6y$$

Add 6y to both sides.

$$-4y-30-y^{2}+6y=2$$

Combine -4y and 6y to get 2y.

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

Add 30 to both sides.

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

Add 2 and 30 to get 32.

$$2y-y^{2}=32$$

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.

$$-y^{2}+2y=32$$

Divide both sides by -1.

$$\frac{-y^{2}+2y}{-1}=\frac{32}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{2}{-1}y=\frac{32}{-1}$$

Divide 2 by -1.

$$y^{2}-2y=\frac{32}{-1}$$

Divide 32 by -1.

$$y^{2}-2y=-32$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-2y+1=-32+1$$

Add -32 to 1.

$$y^{2}-2y+1=-31$$

Factor y^{2}-2y+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-1\right)^{2}=-31$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-1\right)^{2}}=\sqrt{-31}$$

Simplify.

$$y-1=\sqrt{31}i y-1=-\sqrt{31}i$$

Add 1 to both sides of the equation.

$$y=1+\sqrt{31}i y=-\sqrt{31}i+1$$