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x^{2}-9x-5=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(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-5\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\left(-5\right)}}{2}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81+20}}{2}
Multiply -4 times -5.
x=\frac{-\left(-9\right)±\sqrt{101}}{2}
Add 81 to 20.
x=\frac{9±\sqrt{101}}{2}
The opposite of -9 is 9.
x=\frac{\sqrt{101}+9}{2}
Now solve the equation x=\frac{9±\sqrt{101}}{2} when ± is plus. Add 9 to \sqrt{101}.
x=\frac{9-\sqrt{101}}{2}
Now solve the equation x=\frac{9±\sqrt{101}}{2} when ± is minus. Subtract \sqrt{101} from 9.
x=\frac{\sqrt{101}+9}{2} x=\frac{9-\sqrt{101}}{2}
The equation is now solved.
x^{2}-9x-5=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}-9x-5-\left(-5\right)=-\left(-5\right)
Add 5 to both sides of the equation.
x^{2}-9x=-\left(-5\right)
Subtracting -5 from itself leaves 0.
x^{2}-9x=5
Subtract -5 from 0.
x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=5+\left(-\frac{9}{2}\right)^{2}
Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-9x+\frac{81}{4}=5+\frac{81}{4}
Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-9x+\frac{81}{4}=\frac{101}{4}
Add 5 to \frac{81}{4}.
\left(x-\frac{9}{2}\right)^{2}=\frac{101}{4}
Factor x^{2}-9x+\frac{81}{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{9}{2}\right)^{2}}=\sqrt{\frac{101}{4}}
Take the square root of both sides of the equation.
x-\frac{9}{2}=\frac{\sqrt{101}}{2} x-\frac{9}{2}=-\frac{\sqrt{101}}{2}
Simplify.
x=\frac{\sqrt{101}+9}{2} x=\frac{9-\sqrt{101}}{2}
Add \frac{9}{2} to both sides of the equation.