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a+b=25 ab=-12500
To solve the equation, factor x^{2}+25x-12500 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
-1,12500 -2,6250 -4,3125 -5,2500 -10,1250 -20,625 -25,500 -50,250 -100,125
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -12500.
-1+12500=12499 -2+6250=6248 -4+3125=3121 -5+2500=2495 -10+1250=1240 -20+625=605 -25+500=475 -50+250=200 -100+125=25
Calculate the sum for each pair.
a=-100 b=125
The solution is the pair that gives sum 25.
\left(x-100\right)\left(x+125\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=100 x=-125
To find equation solutions, solve x-100=0 and x+125=0.
a+b=25 ab=1\left(-12500\right)=-12500
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-12500. To find a and b, set up a system to be solved.
-1,12500 -2,6250 -4,3125 -5,2500 -10,1250 -20,625 -25,500 -50,250 -100,125
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -12500.
-1+12500=12499 -2+6250=6248 -4+3125=3121 -5+2500=2495 -10+1250=1240 -20+625=605 -25+500=475 -50+250=200 -100+125=25
Calculate the sum for each pair.
a=-100 b=125
The solution is the pair that gives sum 25.
\left(x^{2}-100x\right)+\left(125x-12500\right)
Rewrite x^{2}+25x-12500 as \left(x^{2}-100x\right)+\left(125x-12500\right).
x\left(x-100\right)+125\left(x-100\right)
Factor out x in the first and 125 in the second group.
\left(x-100\right)\left(x+125\right)
Factor out common term x-100 by using distributive property.
x=100 x=-125
To find equation solutions, solve x-100=0 and x+125=0.
x^{2}+25x-12500=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\left(-12500\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 25 for b, and -12500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-25±\sqrt{625-4\left(-12500\right)}}{2}
Square 25.
x=\frac{-25±\sqrt{625+50000}}{2}
Multiply -4 times -12500.
x=\frac{-25±\sqrt{50625}}{2}
Add 625 to 50000.
x=\frac{-25±225}{2}
Take the square root of 50625.
x=\frac{200}{2}
Now solve the equation x=\frac{-25±225}{2} when ± is plus. Add -25 to 225.
x=100
Divide 200 by 2.
x=-\frac{250}{2}
Now solve the equation x=\frac{-25±225}{2} when ± is minus. Subtract 225 from -25.
x=-125
Divide -250 by 2.
x=100 x=-125
The equation is now solved.
x^{2}+25x-12500=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}+25x-12500-\left(-12500\right)=-\left(-12500\right)
Add 12500 to both sides of the equation.
x^{2}+25x=-\left(-12500\right)
Subtracting -12500 from itself leaves 0.
x^{2}+25x=12500
Subtract -12500 from 0.
x^{2}+25x+\left(\frac{25}{2}\right)^{2}=12500+\left(\frac{25}{2}\right)^{2}
Divide 25, the coefficient of the x term, by 2 to get \frac{25}{2}. Then add the square of \frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+25x+\frac{625}{4}=12500+\frac{625}{4}
Square \frac{25}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+25x+\frac{625}{4}=\frac{50625}{4}
Add 12500 to \frac{625}{4}.
\left(x+\frac{25}{2}\right)^{2}=\frac{50625}{4}
Factor x^{2}+25x+\frac{625}{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{25}{2}\right)^{2}}=\sqrt{\frac{50625}{4}}
Take the square root of both sides of the equation.
x+\frac{25}{2}=\frac{225}{2} x+\frac{25}{2}=-\frac{225}{2}
Simplify.
x=100 x=-125
Subtract \frac{25}{2} from both sides of the equation.