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x^{2}-37x+365=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{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 365}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -37 for b, and 365 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-37\right)±\sqrt{1369-4\times 365}}{2}
Square -37.
x=\frac{-\left(-37\right)±\sqrt{1369-1460}}{2}
Multiply -4 times 365.
x=\frac{-\left(-37\right)±\sqrt{-91}}{2}
Add 1369 to -1460.
x=\frac{-\left(-37\right)±\sqrt{91}i}{2}
Take the square root of -91.
x=\frac{37±\sqrt{91}i}{2}
The opposite of -37 is 37.
x=\frac{37+\sqrt{91}i}{2}
Now solve the equation x=\frac{37±\sqrt{91}i}{2} when ± is plus. Add 37 to i\sqrt{91}.
x=\frac{-\sqrt{91}i+37}{2}
Now solve the equation x=\frac{37±\sqrt{91}i}{2} when ± is minus. Subtract i\sqrt{91} from 37.
x=\frac{37+\sqrt{91}i}{2} x=\frac{-\sqrt{91}i+37}{2}
The equation is now solved.
x^{2}-37x+365=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}-37x+365-365=-365
Subtract 365 from both sides of the equation.
x^{2}-37x=-365
Subtracting 365 from itself leaves 0.
x^{2}-37x+\left(-\frac{37}{2}\right)^{2}=-365+\left(-\frac{37}{2}\right)^{2}
Divide -37, the coefficient of the x term, by 2 to get -\frac{37}{2}. Then add the square of -\frac{37}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-37x+\frac{1369}{4}=-365+\frac{1369}{4}
Square -\frac{37}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-37x+\frac{1369}{4}=-\frac{91}{4}
Add -365 to \frac{1369}{4}.
\left(x-\frac{37}{2}\right)^{2}=-\frac{91}{4}
Factor x^{2}-37x+\frac{1369}{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{37}{2}\right)^{2}}=\sqrt{-\frac{91}{4}}
Take the square root of both sides of the equation.
x-\frac{37}{2}=\frac{\sqrt{91}i}{2} x-\frac{37}{2}=-\frac{\sqrt{91}i}{2}
Simplify.
x=\frac{37+\sqrt{91}i}{2} x=\frac{-\sqrt{91}i+37}{2}
Add \frac{37}{2} to both sides of the equation.