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100x^{2}+299x+255=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{-299±\sqrt{299^{2}-4\times 100\times 255}}{2\times 100}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 299 for b, and 255 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-299±\sqrt{89401-4\times 100\times 255}}{2\times 100}
Square 299.
x=\frac{-299±\sqrt{89401-400\times 255}}{2\times 100}
Multiply -4 times 100.
x=\frac{-299±\sqrt{89401-102000}}{2\times 100}
Multiply -400 times 255.
x=\frac{-299±\sqrt{-12599}}{2\times 100}
Add 89401 to -102000.
x=\frac{-299±\sqrt{12599}i}{2\times 100}
Take the square root of -12599.
x=\frac{-299±\sqrt{12599}i}{200}
Multiply 2 times 100.
x=\frac{-299+\sqrt{12599}i}{200}
Now solve the equation x=\frac{-299±\sqrt{12599}i}{200} when ± is plus. Add -299 to i\sqrt{12599}.
x=\frac{-\sqrt{12599}i-299}{200}
Now solve the equation x=\frac{-299±\sqrt{12599}i}{200} when ± is minus. Subtract i\sqrt{12599} from -299.
x=\frac{-299+\sqrt{12599}i}{200} x=\frac{-\sqrt{12599}i-299}{200}
The equation is now solved.
100x^{2}+299x+255=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.
100x^{2}+299x+255-255=-255
Subtract 255 from both sides of the equation.
100x^{2}+299x=-255
Subtracting 255 from itself leaves 0.
\frac{100x^{2}+299x}{100}=-\frac{255}{100}
Divide both sides by 100.
x^{2}+\frac{299}{100}x=-\frac{255}{100}
Dividing by 100 undoes the multiplication by 100.
x^{2}+\frac{299}{100}x=-\frac{51}{20}
Reduce the fraction \frac{-255}{100} to lowest terms by extracting and canceling out 5.
x^{2}+\frac{299}{100}x+\left(\frac{299}{200}\right)^{2}=-\frac{51}{20}+\left(\frac{299}{200}\right)^{2}
Divide \frac{299}{100}, the coefficient of the x term, by 2 to get \frac{299}{200}. Then add the square of \frac{299}{200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{299}{100}x+\frac{89401}{40000}=-\frac{51}{20}+\frac{89401}{40000}
Square \frac{299}{200} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{299}{100}x+\frac{89401}{40000}=-\frac{12599}{40000}
Add -\frac{51}{20} to \frac{89401}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{299}{200}\right)^{2}=-\frac{12599}{40000}
Factor x^{2}+\frac{299}{100}x+\frac{89401}{40000}. 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{299}{200}\right)^{2}}=\sqrt{-\frac{12599}{40000}}
Take the square root of both sides of the equation.
x+\frac{299}{200}=\frac{\sqrt{12599}i}{200} x+\frac{299}{200}=-\frac{\sqrt{12599}i}{200}
Simplify.
x=\frac{-299+\sqrt{12599}i}{200} x=\frac{-\sqrt{12599}i-299}{200}
Subtract \frac{299}{200} from both sides of the equation.