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x\left(x-17\right)=0
Factor out x.
x=0 x=17
To find equation solutions, solve x=0 and x-17=0.
x^{2}-17x=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{-\left(-17\right)±\sqrt{\left(-17\right)^{2}}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -17 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-17\right)±17}{2}
Take the square root of \left(-17\right)^{2}.
x=\frac{17±17}{2}
The opposite of -17 is 17.
x=\frac{34}{2}
Now solve the equation x=\frac{17±17}{2} when ± is plus. Add 17 to 17.
x=17
Divide 34 by 2.
x=\frac{0}{2}
Now solve the equation x=\frac{17±17}{2} when ± is minus. Subtract 17 from 17.
x=0
Divide 0 by 2.
x=17 x=0
The equation is now solved.
x^{2}-17x=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.
x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=\left(-\frac{17}{2}\right)^{2}
Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-17x+\frac{289}{4}=\frac{289}{4}
Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.
\left(x-\frac{17}{2}\right)^{2}=\frac{289}{4}
Factor x^{2}-17x+\frac{289}{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{17}{2}\right)^{2}}=\sqrt{\frac{289}{4}}
Take the square root of both sides of the equation.
x-\frac{17}{2}=\frac{17}{2} x-\frac{17}{2}=-\frac{17}{2}
Simplify.
x=17 x=0
Add \frac{17}{2} to both sides of the equation.