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Solve for x (complex solution)
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x^{2}+2x=-3
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}+2x-\left(-3\right)=-3-\left(-3\right)
Add 3 to both sides of the equation.
x^{2}+2x-\left(-3\right)=0
Subtracting -3 from itself leaves 0.
x^{2}+2x+3=0
Subtract -3 from 0.
x=\frac{-2±\sqrt{2^{2}-4\times 3}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\times 3}}{2}
Square 2.
x=\frac{-2±\sqrt{4-12}}{2}
Multiply -4 times 3.
x=\frac{-2±\sqrt{-8}}{2}
Add 4 to -12.
x=\frac{-2±2\sqrt{2}i}{2}
Take the square root of -8.
x=\frac{-2+2\sqrt{2}i}{2}
Now solve the equation x=\frac{-2±2\sqrt{2}i}{2} when ± is plus. Add -2 to 2i\sqrt{2}.
x=-1+\sqrt{2}i
Divide -2+2i\sqrt{2} by 2.
x=\frac{-2\sqrt{2}i-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{2}i}{2} when ± is minus. Subtract 2i\sqrt{2} from -2.
x=-\sqrt{2}i-1
Divide -2-2i\sqrt{2} by 2.
x=-1+\sqrt{2}i x=-\sqrt{2}i-1
The equation is now solved.
x^{2}+2x=-3
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^{2}=-3+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=-3+1
Square 1.
x^{2}+2x+1=-2
Add -3 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}i x+1=-\sqrt{2}i
Simplify.
x=-1+\sqrt{2}i x=-\sqrt{2}i-1
Subtract 1 from both sides of the equation.