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Solve for x (complex solution)
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x^{2}+2x+3=12
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+3-12=12-12
Subtract 12 from both sides of the equation.
x^{2}+2x+3-12=0
Subtracting 12 from itself leaves 0.
x^{2}+2x-9=0
Subtract 12 from 3.
x=\frac{-2±\sqrt{2^{2}-4\left(-9\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-9\right)}}{2}
Square 2.
x=\frac{-2±\sqrt{4+36}}{2}
Multiply -4 times -9.
x=\frac{-2±\sqrt{40}}{2}
Add 4 to 36.
x=\frac{-2±2\sqrt{10}}{2}
Take the square root of 40.
x=\frac{2\sqrt{10}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{10}}{2} when ± is plus. Add -2 to 2\sqrt{10}.
x=\sqrt{10}-1
Divide -2+2\sqrt{10} by 2.
x=\frac{-2\sqrt{10}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{10}}{2} when ± is minus. Subtract 2\sqrt{10} from -2.
x=-\sqrt{10}-1
Divide -2-2\sqrt{10} by 2.
x=\sqrt{10}-1 x=-\sqrt{10}-1
The equation is now solved.
x^{2}+2x+3=12
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+3-3=12-3
Subtract 3 from both sides of the equation.
x^{2}+2x=12-3
Subtracting 3 from itself leaves 0.
x^{2}+2x=9
Subtract 3 from 12.
x^{2}+2x+1^{2}=9+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=9+1
Square 1.
x^{2}+2x+1=10
Add 9 to 1.
\left(x+1\right)^{2}=10
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{10}
Take the square root of both sides of the equation.
x+1=\sqrt{10} x+1=-\sqrt{10}
Simplify.
x=\sqrt{10}-1 x=-\sqrt{10}-1
Subtract 1 from both sides of the equation.
x^{2}+2x+3=12
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+3-12=12-12
Subtract 12 from both sides of the equation.
x^{2}+2x+3-12=0
Subtracting 12 from itself leaves 0.
x^{2}+2x-9=0
Subtract 12 from 3.
x=\frac{-2±\sqrt{2^{2}-4\left(-9\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-9\right)}}{2}
Square 2.
x=\frac{-2±\sqrt{4+36}}{2}
Multiply -4 times -9.
x=\frac{-2±\sqrt{40}}{2}
Add 4 to 36.
x=\frac{-2±2\sqrt{10}}{2}
Take the square root of 40.
x=\frac{2\sqrt{10}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{10}}{2} when ± is plus. Add -2 to 2\sqrt{10}.
x=\sqrt{10}-1
Divide -2+2\sqrt{10} by 2.
x=\frac{-2\sqrt{10}-2}{2}
Now solve the equation x=\frac{-2±2\sqrt{10}}{2} when ± is minus. Subtract 2\sqrt{10} from -2.
x=-\sqrt{10}-1
Divide -2-2\sqrt{10} by 2.
x=\sqrt{10}-1 x=-\sqrt{10}-1
The equation is now solved.
x^{2}+2x+3=12
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+3-3=12-3
Subtract 3 from both sides of the equation.
x^{2}+2x=12-3
Subtracting 3 from itself leaves 0.
x^{2}+2x=9
Subtract 3 from 12.
x^{2}+2x+1^{2}=9+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=9+1
Square 1.
x^{2}+2x+1=10
Add 9 to 1.
\left(x+1\right)^{2}=10
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{10}
Take the square root of both sides of the equation.
x+1=\sqrt{10} x+1=-\sqrt{10}
Simplify.
x=\sqrt{10}-1 x=-\sqrt{10}-1
Subtract 1 from both sides of the equation.