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x\left(21-x\right)=0
Factor out x.
x=0 x=21
To find equation solutions, solve x=0 and 21-x=0.
-x^{2}+21x=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.
x=\frac{-21±\sqrt{21^{2}}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 21 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-21±21}{2\left(-1\right)}
Take the square root of 21^{2}.
x=\frac{-21±21}{-2}
Multiply 2 times -1.
x=\frac{0}{-2}
Now solve the equation x=\frac{-21±21}{-2} when ± is plus. Add -21 to 21.
x=0
Divide 0 by -2.
x=-\frac{42}{-2}
Now solve the equation x=\frac{-21±21}{-2} when ± is minus. Subtract 21 from -21.
x=21
Divide -42 by -2.
x=0 x=21
The equation is now solved.
-x^{2}+21x=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{-x^{2}+21x}{-1}=\frac{0}{-1}
Divide both sides by -1.
x^{2}+\frac{21}{-1}x=\frac{0}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-21x=\frac{0}{-1}
Divide 21 by -1.
x^{2}-21x=0
Divide 0 by -1.
x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=\left(-\frac{21}{2}\right)^{2}
Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-21x+\frac{441}{4}=\frac{441}{4}
Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.
\left(x-\frac{21}{2}\right)^{2}=\frac{441}{4}
Factor x^{2}-21x+\frac{441}{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(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{441}{4}}
Take the square root of both sides of the equation.
x-\frac{21}{2}=\frac{21}{2} x-\frac{21}{2}=-\frac{21}{2}
Simplify.
x=21 x=0
Add \frac{21}{2} to both sides of the equation.