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z^{2}-z+2=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.
z=\frac{-\left(-1\right)±\sqrt{1-4\times 2}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
z=\frac{-\left(-1\right)±\sqrt{1-8}}{2}
Multiply -4 times 2.
z=\frac{-\left(-1\right)±\sqrt{-7}}{2}
Add 1 to -8.
z=\frac{-\left(-1\right)±\sqrt{7}i}{2}
Take the square root of -7.
z=\frac{1±\sqrt{7}i}{2}
The opposite of -1 is 1.
z=\frac{1+\sqrt{7}i}{2}
Now solve the equation z=\frac{1±\sqrt{7}i}{2} when ± is plus. Add 1 to i\sqrt{7}.
z=\frac{-\sqrt{7}i+1}{2}
Now solve the equation z=\frac{1±\sqrt{7}i}{2} when ± is minus. Subtract i\sqrt{7} from 1.
z=\frac{1+\sqrt{7}i}{2} z=\frac{-\sqrt{7}i+1}{2}
The equation is now solved.
z^{2}-z+2=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.
z^{2}-z+2-2=-2
Subtract 2 from both sides of the equation.
z^{2}-z=-2
Subtracting 2 from itself leaves 0.
z^{2}-z+\left(-\frac{1}{2}\right)^{2}=-2+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
z^{2}-z+\frac{1}{4}=-2+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
z^{2}-z+\frac{1}{4}=-\frac{7}{4}
Add -2 to \frac{1}{4}.
\left(z-\frac{1}{2}\right)^{2}=-\frac{7}{4}
Factor z^{2}-z+\frac{1}{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(z-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}
Take the square root of both sides of the equation.
z-\frac{1}{2}=\frac{\sqrt{7}i}{2} z-\frac{1}{2}=-\frac{\sqrt{7}i}{2}
Simplify.
z=\frac{1+\sqrt{7}i}{2} z=\frac{-\sqrt{7}i+1}{2}
Add \frac{1}{2} to both sides of the equation.