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y\left(7-y\right)=0
Factor out y.
y=0 y=7
To find equation solutions, solve y=0 and 7-y=0.
-y^{2}+7y=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{-7±\sqrt{7^{2}}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-7±7}{2\left(-1\right)}
Take the square root of 7^{2}.
y=\frac{-7±7}{-2}
Multiply 2 times -1.
y=\frac{0}{-2}
Now solve the equation y=\frac{-7±7}{-2} when ± is plus. Add -7 to 7.
y=0
Divide 0 by -2.
y=-\frac{14}{-2}
Now solve the equation y=\frac{-7±7}{-2} when ± is minus. Subtract 7 from -7.
y=7
Divide -14 by -2.
y=0 y=7
The equation is now solved.
-y^{2}+7y=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{-y^{2}+7y}{-1}=\frac{0}{-1}
Divide both sides by -1.
y^{2}+\frac{7}{-1}y=\frac{0}{-1}
Dividing by -1 undoes the multiplication by -1.
y^{2}-7y=\frac{0}{-1}
Divide 7 by -1.
y^{2}-7y=0
Divide 0 by -1.
y^{2}-7y+\left(-\frac{7}{2}\right)^{2}=\left(-\frac{7}{2}\right)^{2}
Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}-7y+\frac{49}{4}=\frac{49}{4}
Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.
\left(y-\frac{7}{2}\right)^{2}=\frac{49}{4}
Factor y^{2}-7y+\frac{49}{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{7}{2}\right)^{2}}=\sqrt{\frac{49}{4}}
Take the square root of both sides of the equation.
y-\frac{7}{2}=\frac{7}{2} y-\frac{7}{2}=-\frac{7}{2}
Simplify.
y=7 y=0
Add \frac{7}{2} to both sides of the equation.