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2x^{2}-x-165=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(-1\right)±\sqrt{1-4\times 2\left(-165\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1 for b, and -165 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1-8\left(-165\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-1\right)±\sqrt{1+1320}}{2\times 2}
Multiply -8 times -165.
x=\frac{-\left(-1\right)±\sqrt{1321}}{2\times 2}
Add 1 to 1320.
x=\frac{1±\sqrt{1321}}{2\times 2}
The opposite of -1 is 1.
x=\frac{1±\sqrt{1321}}{4}
Multiply 2 times 2.
x=\frac{\sqrt{1321}+1}{4}
Now solve the equation x=\frac{1±\sqrt{1321}}{4} when ± is plus. Add 1 to \sqrt{1321}.
x=\frac{1-\sqrt{1321}}{4}
Now solve the equation x=\frac{1±\sqrt{1321}}{4} when ± is minus. Subtract \sqrt{1321} from 1.
x=\frac{\sqrt{1321}+1}{4} x=\frac{1-\sqrt{1321}}{4}
The equation is now solved.
2x^{2}-x-165=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.
2x^{2}-x-165-\left(-165\right)=-\left(-165\right)
Add 165 to both sides of the equation.
2x^{2}-x=-\left(-165\right)
Subtracting -165 from itself leaves 0.
2x^{2}-x=165
Subtract -165 from 0.
\frac{2x^{2}-x}{2}=\frac{165}{2}
Divide both sides by 2.
x^{2}-\frac{1}{2}x=\frac{165}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{165}{2}+\left(-\frac{1}{4}\right)^{2}
Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{165}{2}+\frac{1}{16}
Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1321}{16}
Add \frac{165}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{4}\right)^{2}=\frac{1321}{16}
Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. 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{1}{4}\right)^{2}}=\sqrt{\frac{1321}{16}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{\sqrt{1321}}{4} x-\frac{1}{4}=-\frac{\sqrt{1321}}{4}
Simplify.
x=\frac{\sqrt{1321}+1}{4} x=\frac{1-\sqrt{1321}}{4}
Add \frac{1}{4} to both sides of the equation.