Factor
-\left(x+5\right)\left(x+6\right)
Evaluate
-\left(x+5\right)\left(x+6\right)
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a+b=-11 ab=-\left(-30\right)=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 positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 30.
-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11
Calculate the sum for each pair.
a=-5 b=-6
The solution is the pair that gives sum -11.
\left(-x^{2}-5x\right)+\left(-6x-30\right)
Rewrite -x^{2}-11x-30 as \left(-x^{2}-5x\right)+\left(-6x-30\right).
x\left(-x-5\right)+6\left(-x-5\right)
Factor out x in the first and 6 in the second group.
\left(-x-5\right)\left(x+6\right)
Factor out common term -x-5 by using distributive property.
-x^{2}-11x-30=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(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-30\right)}}{2\left(-1\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(-11\right)±\sqrt{121-4\left(-1\right)\left(-30\right)}}{2\left(-1\right)}
Square -11.
x=\frac{-\left(-11\right)±\sqrt{121+4\left(-30\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-11\right)±\sqrt{121-120}}{2\left(-1\right)}
Multiply 4 times -30.
x=\frac{-\left(-11\right)±\sqrt{1}}{2\left(-1\right)}
Add 121 to -120.
x=\frac{-\left(-11\right)±1}{2\left(-1\right)}
Take the square root of 1.
x=\frac{11±1}{2\left(-1\right)}
The opposite of -11 is 11.
x=\frac{11±1}{-2}
Multiply 2 times -1.
x=\frac{12}{-2}
Now solve the equation x=\frac{11±1}{-2} when ± is plus. Add 11 to 1.
x=-6
Divide 12 by -2.
x=\frac{10}{-2}
Now solve the equation x=\frac{11±1}{-2} when ± is minus. Subtract 1 from 11.
x=-5
Divide 10 by -2.
-x^{2}-11x-30=-\left(x-\left(-6\right)\right)\left(x-\left(-5\right)\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 -5 for x_{2}.
-x^{2}-11x-30=-\left(x+6\right)\left(x+5\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +11x +30 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = -11 rs = 30
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{11}{2} - u s = -\frac{11}{2} + u
Two numbers r and s sum up to -11 exactly when the average of the two numbers is \frac{1}{2}*-11 = -\frac{11}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{11}{2} - u) (-\frac{11}{2} + u) = 30
To solve for unknown quantity u, substitute these in the product equation rs = 30
\frac{121}{4} - u^2 = 30
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 30-\frac{121}{4} = -\frac{1}{4}
Simplify the expression by subtracting \frac{121}{4} on both sides
u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{11}{2} - \frac{1}{2} = -6 s = -\frac{11}{2} + \frac{1}{2} = -5
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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