Solve left(6+4yright)^2-2(6+4y)y=36

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

Solve for y

$$y = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$y=0$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+4y\right)^{2}.

$$36+48y+16y^{2}-2\left(6+4y\right)y=36$$

Subtract 36 from both sides.

$$36+48y+16y^{2}-2\left(6+4y\right)y-36=0$$

Use the distributive property to multiply -2 by 6+4y.

$$36+48y+16y^{2}+\left(-12-8y\right)y-36=0$$

Use the distributive property to multiply -12-8y by y.

$$36+48y+16y^{2}-12y-8y^{2}-36=0$$

Combine 48y and -12y to get 36y.

$$36+36y+16y^{2}-8y^{2}-36=0$$

Combine 16y^{2} and -8y^{2} to get 8y^{2}.

$$36+36y+8y^{2}-36=0$$

Subtract 36 from 36 to get 0.

$$36y+8y^{2}=0$$

Factor out y.

$$y\left(36+8y\right)=0$$

To find equation solutions, solve y=0 and 36+8y=0.

$$y=0 y=-\frac{9}{2}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+4y\right)^{2}.

$$36+48y+16y^{2}-2\left(6+4y\right)y=36$$

Subtract 36 from both sides.

$$36+48y+16y^{2}-2\left(6+4y\right)y-36=0$$

Use the distributive property to multiply -2 by 6+4y.

$$36+48y+16y^{2}+\left(-12-8y\right)y-36=0$$

Use the distributive property to multiply -12-8y by y.

$$36+48y+16y^{2}-12y-8y^{2}-36=0$$

Combine 48y and -12y to get 36y.

$$36+36y+16y^{2}-8y^{2}-36=0$$

Combine 16y^{2} and -8y^{2} to get 8y^{2}.

$$36+36y+8y^{2}-36=0$$

Subtract 36 from 36 to get 0.

$$36y+8y^{2}=0$$

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.

$$8y^{2}+36y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 36 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-36±\sqrt{36^{2}}}{2\times 8}$$

Take the square root of 36^{2}.

$$y=\frac{-36±36}{2\times 8}$$

Multiply 2 times 8.

$$y=\frac{-36±36}{16}$$

Now solve the equation y=\frac{-36±36}{16} when ± is plus. Add -36 to 36.

$$y=\frac{0}{16}$$

Divide 0 by 16.

$$y=0$$

Now solve the equation y=\frac{-36±36}{16} when ± is minus. Subtract 36 from -36.

$$y=-\frac{72}{16}$$

Reduce the fraction \frac{-72}{16} to lowest terms by extracting and canceling out 8.

$$y=-\frac{9}{2}$$

The equation is now solved.

$$y=0 y=-\frac{9}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+4y\right)^{2}.

$$36+48y+16y^{2}-2\left(6+4y\right)y=36$$

Use the distributive property to multiply -2 by 6+4y.

$$36+48y+16y^{2}+\left(-12-8y\right)y=36$$

Use the distributive property to multiply -12-8y by y.

$$36+48y+16y^{2}-12y-8y^{2}=36$$

Combine 48y and -12y to get 36y.

$$36+36y+16y^{2}-8y^{2}=36$$

Combine 16y^{2} and -8y^{2} to get 8y^{2}.

$$36+36y+8y^{2}=36$$

Subtract 36 from both sides.

$$36y+8y^{2}=36-36$$

Subtract 36 from 36 to get 0.

$$36y+8y^{2}=0$$

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.

$$8y^{2}+36y=0$$

Divide both sides by 8.

$$\frac{8y^{2}+36y}{8}=\frac{0}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$y^{2}+\frac{36}{8}y=\frac{0}{8}$$

Reduce the fraction \frac{36}{8} to lowest terms by extracting and canceling out 4.

$$y^{2}+\frac{9}{2}y=\frac{0}{8}$$

Divide 0 by 8.

$$y^{2}+\frac{9}{2}y=0$$

Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{9}{2}y+\left(\frac{9}{4}\right)^{2}=\left(\frac{9}{4}\right)^{2}$$

Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{9}{2}y+\frac{81}{16}=\frac{81}{16}$$

Factor y^{2}+\frac{9}{2}y+\frac{81}{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}.

$$\left(y+\frac{9}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{9}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$y+\frac{9}{4}=\frac{9}{4} y+\frac{9}{4}=-\frac{9}{4}$$

Subtract \frac{9}{4} from both sides of the equation.

$$y=0 y=-\frac{9}{2}$$