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Steps Using the Quadratic Formula
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3\left(-x^{2}-4x+12\right)
Factor out 3.
a+b=-4 ab=-12=-12
Consider -x^{2}-4x+12. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+12. To find a and b, set up a system to be solved.
1,-12 2,-6 3,-4
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 -12.
1-12=-11 2-6=-4 3-4=-1
Calculate the sum for each pair.
a=2 b=-6
The solution is the pair that gives sum -4.
\left(-x^{2}+2x\right)+\left(-6x+12\right)
Rewrite -x^{2}-4x+12 as \left(-x^{2}+2x\right)+\left(-6x+12\right).
x\left(-x+2\right)+6\left(-x+2\right)
Factor out x in the first and 6 in the second group.
\left(-x+2\right)\left(x+6\right)
Factor out common term -x+2 by using distributive property.
3\left(-x+2\right)\left(x+6\right)
Rewrite the complete factored expression.
-3x^{2}-12x+36=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(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 36}}{2\left(-3\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(-12\right)±\sqrt{144-4\left(-3\right)\times 36}}{2\left(-3\right)}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144+12\times 36}}{2\left(-3\right)}
Multiply -4 times -3.
x=\frac{-\left(-12\right)±\sqrt{144+432}}{2\left(-3\right)}
Multiply 12 times 36.
x=\frac{-\left(-12\right)±\sqrt{576}}{2\left(-3\right)}
Add 144 to 432.
x=\frac{-\left(-12\right)±24}{2\left(-3\right)}
Take the square root of 576.
x=\frac{12±24}{2\left(-3\right)}
The opposite of -12 is 12.
x=\frac{12±24}{-6}
Multiply 2 times -3.
x=\frac{36}{-6}
Now solve the equation x=\frac{12±24}{-6} when ± is plus. Add 12 to 24.
x=-6
Divide 36 by -6.
x=\frac{-12}{-6}
Now solve the equation x=\frac{12±24}{-6} when ± is minus. Subtract 24 from 12.
x=2
Divide -12 by -6.
-3x^{2}-12x+36=-3\left(x-\left(-6\right)\right)\left(x-2\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and 2 for x_{2}.
-3x^{2}-12x+36=-3\left(x+6\right)\left(x-2\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.