Solve for x
x=-1
x=3
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a+b=2 ab=-3=-3
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+3. To find a and b, set up a system to be solved.
a=3 b=-1
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.
\left(-x^{2}+3x\right)+\left(-x+3\right)
Rewrite -x^{2}+2x+3 as \left(-x^{2}+3x\right)+\left(-x+3\right).
-x\left(x-3\right)-\left(x-3\right)
Factor out -x in the first and -1 in the second group.
\left(x-3\right)\left(-x-1\right)
Factor out common term x-3 by using distributive property.
x=3 x=-1
To find equation solutions, solve x-3=0 and -x-1=0.
-x^{2}+2x+3=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(-1\right)\times 3}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-1\right)\times 3}}{2\left(-1\right)}
Square 2.
x=\frac{-2±\sqrt{4+4\times 3}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-2±\sqrt{4+12}}{2\left(-1\right)}
Multiply 4 times 3.
x=\frac{-2±\sqrt{16}}{2\left(-1\right)}
Add 4 to 12.
x=\frac{-2±4}{2\left(-1\right)}
Take the square root of 16.
x=\frac{-2±4}{-2}
Multiply 2 times -1.
x=\frac{2}{-2}
Now solve the equation x=\frac{-2±4}{-2} when ± is plus. Add -2 to 4.
x=-1
Divide 2 by -2.
x=-\frac{6}{-2}
Now solve the equation x=\frac{-2±4}{-2} when ± is minus. Subtract 4 from -2.
x=3
Divide -6 by -2.
x=-1 x=3
The equation is now solved.
-x^{2}+2x+3=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+3-3=-3
Subtract 3 from both sides of the equation.
-x^{2}+2x=-3
Subtracting 3 from itself leaves 0.
\frac{-x^{2}+2x}{-1}=-\frac{3}{-1}
Divide both sides by -1.
x^{2}+\frac{2}{-1}x=-\frac{3}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-2x=-\frac{3}{-1}
Divide 2 by -1.
x^{2}-2x=3
Divide -3 by -1.
x^{2}-2x+1=3+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=4
Add 3 to 1.
\left(x-1\right)^{2}=4
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{4}
Take the square root of both sides of the equation.
x-1=2 x-1=-2
Simplify.
x=3 x=-1
Add 1 to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}