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