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2522x^{2}+25x-9=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{-25±\sqrt{25^{2}-4\times 2522\left(-9\right)}}{2\times 2522}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2522 for a, 25 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-25±\sqrt{625-4\times 2522\left(-9\right)}}{2\times 2522}
Square 25.
x=\frac{-25±\sqrt{625-10088\left(-9\right)}}{2\times 2522}
Multiply -4 times 2522.
x=\frac{-25±\sqrt{625+90792}}{2\times 2522}
Multiply -10088 times -9.
x=\frac{-25±\sqrt{91417}}{2\times 2522}
Add 625 to 90792.
x=\frac{-25±\sqrt{91417}}{5044}
Multiply 2 times 2522.
x=\frac{\sqrt{91417}-25}{5044}
Now solve the equation x=\frac{-25±\sqrt{91417}}{5044} when ± is plus. Add -25 to \sqrt{91417}.
x=\frac{-\sqrt{91417}-25}{5044}
Now solve the equation x=\frac{-25±\sqrt{91417}}{5044} when ± is minus. Subtract \sqrt{91417} from -25.
x=\frac{\sqrt{91417}-25}{5044} x=\frac{-\sqrt{91417}-25}{5044}
The equation is now solved.
2522x^{2}+25x-9=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.
2522x^{2}+25x-9-\left(-9\right)=-\left(-9\right)
Add 9 to both sides of the equation.
2522x^{2}+25x=-\left(-9\right)
Subtracting -9 from itself leaves 0.
2522x^{2}+25x=9
Subtract -9 from 0.
\frac{2522x^{2}+25x}{2522}=\frac{9}{2522}
Divide both sides by 2522.
x^{2}+\frac{25}{2522}x=\frac{9}{2522}
Dividing by 2522 undoes the multiplication by 2522.
x^{2}+\frac{25}{2522}x+\left(\frac{25}{5044}\right)^{2}=\frac{9}{2522}+\left(\frac{25}{5044}\right)^{2}
Divide \frac{25}{2522}, the coefficient of the x term, by 2 to get \frac{25}{5044}. Then add the square of \frac{25}{5044} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{25}{2522}x+\frac{625}{25441936}=\frac{9}{2522}+\frac{625}{25441936}
Square \frac{25}{5044} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{25}{2522}x+\frac{625}{25441936}=\frac{91417}{25441936}
Add \frac{9}{2522} to \frac{625}{25441936} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{25}{5044}\right)^{2}=\frac{91417}{25441936}
Factor x^{2}+\frac{25}{2522}x+\frac{625}{25441936}. 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{25}{5044}\right)^{2}}=\sqrt{\frac{91417}{25441936}}
Take the square root of both sides of the equation.
x+\frac{25}{5044}=\frac{\sqrt{91417}}{5044} x+\frac{25}{5044}=-\frac{\sqrt{91417}}{5044}
Simplify.
x=\frac{\sqrt{91417}-25}{5044} x=\frac{-\sqrt{91417}-25}{5044}
Subtract \frac{25}{5044} from both sides of the equation.