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x^{2}-9x+144=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\times 144}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\times 144}}{2}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81-576}}{2}
Multiply -4 times 144.
x=\frac{-\left(-9\right)±\sqrt{-495}}{2}
Add 81 to -576.
x=\frac{-\left(-9\right)±3\sqrt{55}i}{2}
Take the square root of -495.
x=\frac{9±3\sqrt{55}i}{2}
The opposite of -9 is 9.
x=\frac{9+3\sqrt{55}i}{2}
Now solve the equation x=\frac{9±3\sqrt{55}i}{2} when ± is plus. Add 9 to 3i\sqrt{55}.
x=\frac{-3\sqrt{55}i+9}{2}
Now solve the equation x=\frac{9±3\sqrt{55}i}{2} when ± is minus. Subtract 3i\sqrt{55} from 9.
x=\frac{9+3\sqrt{55}i}{2} x=\frac{-3\sqrt{55}i+9}{2}
The equation is now solved.
x^{2}-9x+144=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+144-144=-144
Subtract 144 from both sides of the equation.
x^{2}-9x=-144
Subtracting 144 from itself leaves 0.
x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-144+\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}=-144+\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{495}{4}
Add -144 to \frac{81}{4}.
\left(x-\frac{9}{2}\right)^{2}=-\frac{495}{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{495}{4}}
Take the square root of both sides of the equation.
x-\frac{9}{2}=\frac{3\sqrt{55}i}{2} x-\frac{9}{2}=-\frac{3\sqrt{55}i}{2}
Simplify.
x=\frac{9+3\sqrt{55}i}{2} x=\frac{-3\sqrt{55}i+9}{2}
Add \frac{9}{2} to both sides of the equation.