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Solve for x (complex solution)
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-2=x^{2}-4x+3
Use the distributive property to multiply x-3 by x-1 and combine like terms.
x^{2}-4x+3=-2
Swap sides so that all variable terms are on the left hand side.
x^{2}-4x+3+2=0
Add 2 to both sides.
x^{2}-4x+5=0
Add 3 and 2 to get 5.
x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 5}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\times 5}}{2}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16-20}}{2}
Multiply -4 times 5.
x=\frac{-\left(-4\right)±\sqrt{-4}}{2}
Add 16 to -20.
x=\frac{-\left(-4\right)±2i}{2}
Take the square root of -4.
x=\frac{4±2i}{2}
The opposite of -4 is 4.
x=\frac{4+2i}{2}
Now solve the equation x=\frac{4±2i}{2} when ± is plus. Add 4 to 2i.
x=2+i
Divide 4+2i by 2.
x=\frac{4-2i}{2}
Now solve the equation x=\frac{4±2i}{2} when ± is minus. Subtract 2i from 4.
x=2-i
Divide 4-2i by 2.
x=2+i x=2-i
The equation is now solved.
-2=x^{2}-4x+3
Use the distributive property to multiply x-3 by x-1 and combine like terms.
x^{2}-4x+3=-2
Swap sides so that all variable terms are on the left hand side.
x^{2}-4x=-2-3
Subtract 3 from both sides.
x^{2}-4x=-5
Subtract 3 from -2 to get -5.
x^{2}-4x+\left(-2\right)^{2}=-5+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=-5+4
Square -2.
x^{2}-4x+4=-1
Add -5 to 4.
\left(x-2\right)^{2}=-1
Factor x^{2}-4x+4. 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-2\right)^{2}}=\sqrt{-1}
Take the square root of both sides of the equation.
x-2=i x-2=-i
Simplify.
x=2+i x=2-i
Add 2 to both sides of the equation.