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x^{2}-7x-2=\frac{1}{2}
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^{2}-7x-2-\frac{1}{2}=\frac{1}{2}-\frac{1}{2}
Subtract \frac{1}{2} from both sides of the equation.
x^{2}-7x-2-\frac{1}{2}=0
Subtracting \frac{1}{2} from itself leaves 0.
x^{2}-7x-\frac{5}{2}=0
Subtract \frac{1}{2} from -2.
x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-\frac{5}{2}\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and -\frac{5}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-7\right)±\sqrt{49-4\left(-\frac{5}{2}\right)}}{2}
Square -7.
x=\frac{-\left(-7\right)±\sqrt{49+10}}{2}
Multiply -4 times -\frac{5}{2}.
x=\frac{-\left(-7\right)±\sqrt{59}}{2}
Add 49 to 10.
x=\frac{7±\sqrt{59}}{2}
The opposite of -7 is 7.
x=\frac{\sqrt{59}+7}{2}
Now solve the equation x=\frac{7±\sqrt{59}}{2} when ± is plus. Add 7 to \sqrt{59}.
x=\frac{7-\sqrt{59}}{2}
Now solve the equation x=\frac{7±\sqrt{59}}{2} when ± is minus. Subtract \sqrt{59} from 7.
x=\frac{\sqrt{59}+7}{2} x=\frac{7-\sqrt{59}}{2}
The equation is now solved.
x^{2}-7x-2=\frac{1}{2}
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}-7x-2-\left(-2\right)=\frac{1}{2}-\left(-2\right)
Add 2 to both sides of the equation.
x^{2}-7x=\frac{1}{2}-\left(-2\right)
Subtracting -2 from itself leaves 0.
x^{2}-7x=\frac{5}{2}
Subtract -2 from \frac{1}{2}.
x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=\frac{5}{2}+\left(-\frac{7}{2}\right)^{2}
Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-7x+\frac{49}{4}=\frac{5}{2}+\frac{49}{4}
Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-7x+\frac{49}{4}=\frac{59}{4}
Add \frac{5}{2} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{7}{2}\right)^{2}=\frac{59}{4}
Factor x^{2}-7x+\frac{49}{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{7}{2}\right)^{2}}=\sqrt{\frac{59}{4}}
Take the square root of both sides of the equation.
x-\frac{7}{2}=\frac{\sqrt{59}}{2} x-\frac{7}{2}=-\frac{\sqrt{59}}{2}
Simplify.
x=\frac{\sqrt{59}+7}{2} x=\frac{7-\sqrt{59}}{2}
Add \frac{7}{2} to both sides of the equation.