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x^{2}+401x-12132=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{-401±\sqrt{401^{2}-4\left(-12132\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 401 for b, and -12132 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-401±\sqrt{160801-4\left(-12132\right)}}{2}
Square 401.
x=\frac{-401±\sqrt{160801+48528}}{2}
Multiply -4 times -12132.
x=\frac{-401±\sqrt{209329}}{2}
Add 160801 to 48528.
x=\frac{\sqrt{209329}-401}{2}
Now solve the equation x=\frac{-401±\sqrt{209329}}{2} when ± is plus. Add -401 to \sqrt{209329}.
x=\frac{-\sqrt{209329}-401}{2}
Now solve the equation x=\frac{-401±\sqrt{209329}}{2} when ± is minus. Subtract \sqrt{209329} from -401.
x=\frac{\sqrt{209329}-401}{2} x=\frac{-\sqrt{209329}-401}{2}
The equation is now solved.
x^{2}+401x-12132=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}+401x-12132-\left(-12132\right)=-\left(-12132\right)
Add 12132 to both sides of the equation.
x^{2}+401x=-\left(-12132\right)
Subtracting -12132 from itself leaves 0.
x^{2}+401x=12132
Subtract -12132 from 0.
x^{2}+401x+\left(\frac{401}{2}\right)^{2}=12132+\left(\frac{401}{2}\right)^{2}
Divide 401, the coefficient of the x term, by 2 to get \frac{401}{2}. Then add the square of \frac{401}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+401x+\frac{160801}{4}=12132+\frac{160801}{4}
Square \frac{401}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+401x+\frac{160801}{4}=\frac{209329}{4}
Add 12132 to \frac{160801}{4}.
\left(x+\frac{401}{2}\right)^{2}=\frac{209329}{4}
Factor x^{2}+401x+\frac{160801}{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{401}{2}\right)^{2}}=\sqrt{\frac{209329}{4}}
Take the square root of both sides of the equation.
x+\frac{401}{2}=\frac{\sqrt{209329}}{2} x+\frac{401}{2}=-\frac{\sqrt{209329}}{2}
Simplify.
x=\frac{\sqrt{209329}-401}{2} x=\frac{-\sqrt{209329}-401}{2}
Subtract \frac{401}{2} from both sides of the equation.