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Steps Using the Quadratic Formula
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2\left(-x^{2}+x+30\right)
Factor out 2.
a+b=1 ab=-30=-30
Consider -x^{2}+x+30. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+30. To find a and b, set up a system to be solved.
-1,30 -2,15 -3,10 -5,6
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. List all such integer pairs that give product -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=6 b=-5
The solution is the pair that gives sum 1.
\left(-x^{2}+6x\right)+\left(-5x+30\right)
Rewrite -x^{2}+x+30 as \left(-x^{2}+6x\right)+\left(-5x+30\right).
-x\left(x-6\right)-5\left(x-6\right)
Factor out -x in the first and -5 in the second group.
\left(x-6\right)\left(-x-5\right)
Factor out common term x-6 by using distributive property.
2\left(x-6\right)\left(-x-5\right)
Rewrite the complete factored expression.
-2x^{2}+2x+60=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{-2±\sqrt{2^{2}-4\left(-2\right)\times 60}}{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{-2±\sqrt{4-4\left(-2\right)\times 60}}{2\left(-2\right)}
Square 2.
x=\frac{-2±\sqrt{4+8\times 60}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-2±\sqrt{4+480}}{2\left(-2\right)}
Multiply 8 times 60.
x=\frac{-2±\sqrt{484}}{2\left(-2\right)}
Add 4 to 480.
x=\frac{-2±22}{2\left(-2\right)}
Take the square root of 484.
x=\frac{-2±22}{-4}
Multiply 2 times -2.
x=\frac{20}{-4}
Now solve the equation x=\frac{-2±22}{-4} when ± is plus. Add -2 to 22.
x=-5
Divide 20 by -4.
x=\frac{-24}{-4}
Now solve the equation x=\frac{-2±22}{-4} when ± is minus. Subtract 22 from -2.
x=6
Divide -24 by -4.
-2x^{2}+2x+60=-2\left(x-\left(-5\right)\right)\left(x-6\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and 6 for x_{2}.
-2x^{2}+2x+60=-2\left(x+5\right)\left(x-6\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.