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Solve for x (complex solution)
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x^{2}-2x+40=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{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 40}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4\times 40}}{2}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{4-160}}{2}
Multiply -4 times 40.
x=\frac{-\left(-2\right)±\sqrt{-156}}{2}
Add 4 to -160.
x=\frac{-\left(-2\right)±2\sqrt{39}i}{2}
Take the square root of -156.
x=\frac{2±2\sqrt{39}i}{2}
The opposite of -2 is 2.
x=\frac{2+2\sqrt{39}i}{2}
Now solve the equation x=\frac{2±2\sqrt{39}i}{2} when ± is plus. Add 2 to 2i\sqrt{39}.
x=1+\sqrt{39}i
Divide 2+2i\sqrt{39} by 2.
x=\frac{-2\sqrt{39}i+2}{2}
Now solve the equation x=\frac{2±2\sqrt{39}i}{2} when ± is minus. Subtract 2i\sqrt{39} from 2.
x=-\sqrt{39}i+1
Divide 2-2i\sqrt{39} by 2.
x=1+\sqrt{39}i x=-\sqrt{39}i+1
The equation is now solved.
x^{2}-2x+40=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+40-40=-40
Subtract 40 from both sides of the equation.
x^{2}-2x=-40
Subtracting 40 from itself leaves 0.
x^{2}-2x+1=-40+1
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=-39
Add -40 to 1.
\left(x-1\right)^{2}=-39
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{-39}
Take the square root of both sides of the equation.
x-1=\sqrt{39}i x-1=-\sqrt{39}i
Simplify.
x=1+\sqrt{39}i x=-\sqrt{39}i+1
Add 1 to both sides of the equation.