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Solve for x (complex solution)
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x^{2}+2x-12=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(-12\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-12\right)}}{2}
Square 2.
x=\frac{-2±\sqrt{4+48}}{2}
Multiply -4 times -12.
x=\frac{-2±\sqrt{52}}{2}
Add 4 to 48.
x=\frac{-2±2\sqrt{13}}{2}
Take the square root of 52.
x=\frac{2\sqrt{13}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{13}}{2} when ± is plus. Add -2 to 2\sqrt{13}.
x=\sqrt{13}-1
Divide -2+2\sqrt{13} by 2.
x=\frac{-2\sqrt{13}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{13}}{2} when ± is minus. Subtract 2\sqrt{13} from -2.
x=-\sqrt{13}-1
Divide -2-2\sqrt{13} by 2.
x=\sqrt{13}-1 x=-\sqrt{13}-1
The equation is now solved.
x^{2}+2x-12=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-12-\left(-12\right)=-\left(-12\right)
Add 12 to both sides of the equation.
x^{2}+2x=-\left(-12\right)
Subtracting -12 from itself leaves 0.
x^{2}+2x=12
Subtract -12 from 0.
x^{2}+2x+1^{2}=12+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=12+1
Square 1.
x^{2}+2x+1=13
Add 12 to 1.
\left(x+1\right)^{2}=13
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{13}
Take the square root of both sides of the equation.
x+1=\sqrt{13} x+1=-\sqrt{13}
Simplify.
x=\sqrt{13}-1 x=-\sqrt{13}-1
Subtract 1 from both sides of the equation.
x^{2}+2x-12=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(-12\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-12\right)}}{2}
Square 2.
x=\frac{-2±\sqrt{4+48}}{2}
Multiply -4 times -12.
x=\frac{-2±\sqrt{52}}{2}
Add 4 to 48.
x=\frac{-2±2\sqrt{13}}{2}
Take the square root of 52.
x=\frac{2\sqrt{13}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{13}}{2} when ± is plus. Add -2 to 2\sqrt{13}.
x=\sqrt{13}-1
Divide -2+2\sqrt{13} by 2.
x=\frac{-2\sqrt{13}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{13}}{2} when ± is minus. Subtract 2\sqrt{13} from -2.
x=-\sqrt{13}-1
Divide -2-2\sqrt{13} by 2.
x=\sqrt{13}-1 x=-\sqrt{13}-1
The equation is now solved.
x^{2}+2x-12=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-12-\left(-12\right)=-\left(-12\right)
Add 12 to both sides of the equation.
x^{2}+2x=-\left(-12\right)
Subtracting -12 from itself leaves 0.
x^{2}+2x=12
Subtract -12 from 0.
x^{2}+2x+1^{2}=12+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=12+1
Square 1.
x^{2}+2x+1=13
Add 12 to 1.
\left(x+1\right)^{2}=13
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{13}
Take the square root of both sides of the equation.
x+1=\sqrt{13} x+1=-\sqrt{13}
Simplify.
x=\sqrt{13}-1 x=-\sqrt{13}-1
Subtract 1 from both sides of the equation.