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