Solve 5y+9y^2-4y^2+6(5y+9)y=-12

$$5 y + 9 y ^ { 2 } - 4 y ^ { 2 } + 6 ( 5 y + 9 ) y = - 12$$

Solve for y

$$y=\frac{\sqrt{1801}-59}{70}\approx -0.236597281$$

$$y=\frac{-\sqrt{1801}-59}{70}\approx -1.449117005$$

Steps Using the Quadratic Formula

Combine 9y^{2} and -4y^{2} to get 5y^{2}.

$$5y+5y^{2}+6\left(5y+9\right)y=-12$$

Use the distributive property to multiply 6 by 5y+9.

$$5y+5y^{2}+\left(30y+54\right)y=-12$$

Use the distributive property to multiply 30y+54 by y.

$$5y+5y^{2}+30y^{2}+54y=-12$$

Combine 5y^{2} and 30y^{2} to get 35y^{2}.

$$5y+35y^{2}+54y=-12$$

Combine 5y and 54y to get 59y.

$$59y+35y^{2}=-12$$

Add 12 to both sides.

$$59y+35y^{2}+12=0$$

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.

$$35y^{2}+59y+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, 59 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-59±\sqrt{59^{2}-4\times 35\times 12}}{2\times 35}$$

Square 59.

$$y=\frac{-59±\sqrt{3481-4\times 35\times 12}}{2\times 35}$$

Multiply -4 times 35.

$$y=\frac{-59±\sqrt{3481-140\times 12}}{2\times 35}$$

Multiply -140 times 12.

$$y=\frac{-59±\sqrt{3481-1680}}{2\times 35}$$

Add 3481 to -1680.

$$y=\frac{-59±\sqrt{1801}}{2\times 35}$$

Multiply 2 times 35.

$$y=\frac{-59±\sqrt{1801}}{70}$$

Now solve the equation y=\frac{-59±\sqrt{1801}}{70} when ± is plus. Add -59 to \sqrt{1801}.

$$y=\frac{\sqrt{1801}-59}{70}$$

Now solve the equation y=\frac{-59±\sqrt{1801}}{70} when ± is minus. Subtract \sqrt{1801} from -59.

$$y=\frac{-\sqrt{1801}-59}{70}$$

The equation is now solved.

$$y=\frac{\sqrt{1801}-59}{70} y=\frac{-\sqrt{1801}-59}{70}$$

Steps for Completing the Square

Combine 9y^{2} and -4y^{2} to get 5y^{2}.

$$5y+5y^{2}+6\left(5y+9\right)y=-12$$

Use the distributive property to multiply 6 by 5y+9.

$$5y+5y^{2}+\left(30y+54\right)y=-12$$

Use the distributive property to multiply 30y+54 by y.

$$5y+5y^{2}+30y^{2}+54y=-12$$

Combine 5y^{2} and 30y^{2} to get 35y^{2}.

$$5y+35y^{2}+54y=-12$$

Combine 5y and 54y to get 59y.

$$59y+35y^{2}=-12$$

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.

$$35y^{2}+59y=-12$$

Divide both sides by 35.

$$\frac{35y^{2}+59y}{35}=-\frac{12}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$y^{2}+\frac{59}{35}y=-\frac{12}{35}$$

Divide \frac{59}{35}, the coefficient of the x term, by 2 to get \frac{59}{70}. Then add the square of \frac{59}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{59}{35}y+\left(\frac{59}{70}\right)^{2}=-\frac{12}{35}+\left(\frac{59}{70}\right)^{2}$$

Square \frac{59}{70} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{59}{35}y+\frac{3481}{4900}=-\frac{12}{35}+\frac{3481}{4900}$$

Add -\frac{12}{35} to \frac{3481}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{59}{35}y+\frac{3481}{4900}=\frac{1801}{4900}$$

Factor y^{2}+\frac{59}{35}y+\frac{3481}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{59}{70}\right)^{2}=\frac{1801}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{59}{70}\right)^{2}}=\sqrt{\frac{1801}{4900}}$$

Simplify.

$$y+\frac{59}{70}=\frac{\sqrt{1801}}{70} y+\frac{59}{70}=-\frac{\sqrt{1801}}{70}$$

Subtract \frac{59}{70} from both sides of the equation.

$$y=\frac{\sqrt{1801}-59}{70} y=\frac{-\sqrt{1801}-59}{70}$$