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a+b=-3 ab=-2\times 9=-18
Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+9. To find a and b, set up a system to be solved.
1,-18 2,-9 3,-6
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -18.
1-18=-17 2-9=-7 3-6=-3
Calculate the sum for each pair.
a=3 b=-6
The solution is the pair that gives sum -3.
\left(-2x^{2}+3x\right)+\left(-6x+9\right)
Rewrite -2x^{2}-3x+9 as \left(-2x^{2}+3x\right)+\left(-6x+9\right).
-x\left(2x-3\right)-3\left(2x-3\right)
Factor out -x in the first and -3 in the second group.
\left(2x-3\right)\left(-x-3\right)
Factor out common term 2x-3 by using distributive property.
-2x^{2}-3x+9=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-2\right)\times 9}}{2\left(-2\right)}
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(-3\right)±\sqrt{9-4\left(-2\right)\times 9}}{2\left(-2\right)}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9+8\times 9}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-\left(-3\right)±\sqrt{9+72}}{2\left(-2\right)}
Multiply 8 times 9.
x=\frac{-\left(-3\right)±\sqrt{81}}{2\left(-2\right)}
Add 9 to 72.
x=\frac{-\left(-3\right)±9}{2\left(-2\right)}
Take the square root of 81.
x=\frac{3±9}{2\left(-2\right)}
The opposite of -3 is 3.
x=\frac{3±9}{-4}
Multiply 2 times -2.
x=\frac{12}{-4}
Now solve the equation x=\frac{3±9}{-4} when ± is plus. Add 3 to 9.
x=-3
Divide 12 by -4.
x=-\frac{6}{-4}
Now solve the equation x=\frac{3±9}{-4} when ± is minus. Subtract 9 from 3.
x=\frac{3}{2}
Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.
-2x^{2}-3x+9=-2\left(x-\left(-3\right)\right)\left(x-\frac{3}{2}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and \frac{3}{2} for x_{2}.
-2x^{2}-3x+9=-2\left(x+3\right)\left(x-\frac{3}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
-2x^{2}-3x+9=-2\left(x+3\right)\times \frac{-2x+3}{-2}
Subtract \frac{3}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
-2x^{2}-3x+9=\left(x+3\right)\left(-2x+3\right)
Cancel out 2, the greatest common factor in -2 and 2.