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c^{2}-10c-125=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.
c=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-125\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
c=\frac{-\left(-10\right)±\sqrt{100-4\left(-125\right)}}{2}
Square -10.
c=\frac{-\left(-10\right)±\sqrt{100+500}}{2}
Multiply -4 times -125.
c=\frac{-\left(-10\right)±\sqrt{600}}{2}
Add 100 to 500.
c=\frac{-\left(-10\right)±10\sqrt{6}}{2}
Take the square root of 600.
c=\frac{10±10\sqrt{6}}{2}
The opposite of -10 is 10.
c=\frac{10\sqrt{6}+10}{2}
Now solve the equation c=\frac{10±10\sqrt{6}}{2} when ± is plus. Add 10 to 10\sqrt{6}.
c=5\sqrt{6}+5
Divide 10+10\sqrt{6} by 2.
c=\frac{10-10\sqrt{6}}{2}
Now solve the equation c=\frac{10±10\sqrt{6}}{2} when ± is minus. Subtract 10\sqrt{6} from 10.
c=5-5\sqrt{6}
Divide 10-10\sqrt{6} by 2.
c=5\sqrt{6}+5 c=5-5\sqrt{6}
The equation is now solved.
c^{2}-10c-125=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.
c^{2}-10c-125-\left(-125\right)=-\left(-125\right)
Add 125 to both sides of the equation.
c^{2}-10c=-\left(-125\right)
Subtracting -125 from itself leaves 0.
c^{2}-10c=125
Subtract -125 from 0.
c^{2}-10c+\left(-5\right)^{2}=125+\left(-5\right)^{2}
Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
c^{2}-10c+25=125+25
Square -5.
c^{2}-10c+25=150
Add 125 to 25.
\left(c-5\right)^{2}=150
Factor c^{2}-10c+25. 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(c-5\right)^{2}}=\sqrt{150}
Take the square root of both sides of the equation.
c-5=5\sqrt{6} c-5=-5\sqrt{6}
Simplify.
c=5\sqrt{6}+5 c=5-5\sqrt{6}
Add 5 to both sides of the equation.