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y\left(5y+16\right)=0
Factor out y.
y=0 y=-\frac{16}{5}
To find equation solutions, solve y=0 and 5y+16=0.
5y^{2}+16y=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{-16±\sqrt{16^{2}}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 16 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-16±16}{2\times 5}
Take the square root of 16^{2}.
y=\frac{-16±16}{10}
Multiply 2 times 5.
y=\frac{0}{10}
Now solve the equation y=\frac{-16±16}{10} when ± is plus. Add -16 to 16.
y=0
Divide 0 by 10.
y=-\frac{32}{10}
Now solve the equation y=\frac{-16±16}{10} when ± is minus. Subtract 16 from -16.
y=-\frac{16}{5}
Reduce the fraction \frac{-32}{10} to lowest terms by extracting and canceling out 2.
y=0 y=-\frac{16}{5}
The equation is now solved.
5y^{2}+16y=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.
\frac{5y^{2}+16y}{5}=\frac{0}{5}
Divide both sides by 5.
y^{2}+\frac{16}{5}y=\frac{0}{5}
Dividing by 5 undoes the multiplication by 5.
y^{2}+\frac{16}{5}y=0
Divide 0 by 5.
y^{2}+\frac{16}{5}y+\left(\frac{8}{5}\right)^{2}=\left(\frac{8}{5}\right)^{2}
Divide \frac{16}{5}, the coefficient of the x term, by 2 to get \frac{8}{5}. Then add the square of \frac{8}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}+\frac{16}{5}y+\frac{64}{25}=\frac{64}{25}
Square \frac{8}{5} by squaring both the numerator and the denominator of the fraction.
\left(y+\frac{8}{5}\right)^{2}=\frac{64}{25}
Factor y^{2}+\frac{16}{5}y+\frac{64}{25}. 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{8}{5}\right)^{2}}=\sqrt{\frac{64}{25}}
Take the square root of both sides of the equation.
y+\frac{8}{5}=\frac{8}{5} y+\frac{8}{5}=-\frac{8}{5}
Simplify.
y=0 y=-\frac{16}{5}
Subtract \frac{8}{5} from both sides of the equation.