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