Solve y^2+9=-9y

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

Solve for y

$$y=\frac{3\sqrt{5}-9}{2}\approx -1.145898034$$

$$y=\frac{-3\sqrt{5}-9}{2}\approx -7.854101966$$

Steps Using the Quadratic Formula

Add 9y to both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-9±\sqrt{9^{2}-4\times 9}}{2}$$

Square 9.

$$y=\frac{-9±\sqrt{81-4\times 9}}{2}$$

Multiply -4 times 9.

$$y=\frac{-9±\sqrt{81-36}}{2}$$

Add 81 to -36.

$$y=\frac{-9±\sqrt{45}}{2}$$

Take the square root of 45.

$$y=\frac{-9±3\sqrt{5}}{2}$$

Now solve the equation y=\frac{-9±3\sqrt{5}}{2} when ± is plus. Add -9 to 3\sqrt{5}.

$$y=\frac{3\sqrt{5}-9}{2}$$

Now solve the equation y=\frac{-9±3\sqrt{5}}{2} when ± is minus. Subtract 3\sqrt{5} from -9.

$$y=\frac{-3\sqrt{5}-9}{2}$$

The equation is now solved.

$$y=\frac{3\sqrt{5}-9}{2} y=\frac{-3\sqrt{5}-9}{2}$$

Steps for Completing the Square

Add 9y to both sides.

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

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

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

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

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

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+9y+\frac{81}{4}=-9+\frac{81}{4}$$

Add -9 to \frac{81}{4}.

$$y^{2}+9y+\frac{81}{4}=\frac{45}{4}$$

Factor y^{2}+9y+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{9}{2}\right)^{2}=\frac{45}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{9}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Simplify.

$$y+\frac{9}{2}=\frac{3\sqrt{5}}{2} y+\frac{9}{2}=-\frac{3\sqrt{5}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$y=\frac{3\sqrt{5}-9}{2} y=\frac{-3\sqrt{5}-9}{2}$$