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y\left(y+33\right)=0
Factor out y.
y=0 y=-33
To find equation solutions, solve y=0 and y+33=0.
y^{2}+33y=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=\frac{-33±\sqrt{33^{2}}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 33 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-33±33}{2}
Take the square root of 33^{2}.
y=\frac{0}{2}
Now solve the equation y=\frac{-33±33}{2} when ± is plus. Add -33 to 33.
y=0
Divide 0 by 2.
y=-\frac{66}{2}
Now solve the equation y=\frac{-33±33}{2} when ± is minus. Subtract 33 from -33.
y=-33
Divide -66 by 2.
y=0 y=-33
The equation is now solved.
y^{2}+33y=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.
y^{2}+33y+\left(\frac{33}{2}\right)^{2}=\left(\frac{33}{2}\right)^{2}
Divide 33, the coefficient of the x term, by 2 to get \frac{33}{2}. Then add the square of \frac{33}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}+33y+\frac{1089}{4}=\frac{1089}{4}
Square \frac{33}{2} by squaring both the numerator and the denominator of the fraction.
\left(y+\frac{33}{2}\right)^{2}=\frac{1089}{4}
Factor y^{2}+33y+\frac{1089}{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}.
\sqrt{\left(y+\frac{33}{2}\right)^{2}}=\sqrt{\frac{1089}{4}}
Take the square root of both sides of the equation.
y+\frac{33}{2}=\frac{33}{2} y+\frac{33}{2}=-\frac{33}{2}
Simplify.
y=0 y=-33
Subtract \frac{33}{2} from both sides of the equation.