Factor
\left(x-12\right)\left(x+3\right)
Evaluate
\left(x-12\right)\left(x+3\right)
Graph
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a+b=-9 ab=1\left(-36\right)=-36
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-36. To find a and b, set up a system to be solved.
1,-36 2,-18 3,-12 4,-9 6,-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 -36.
1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0
Calculate the sum for each pair.
a=-12 b=3
The solution is the pair that gives sum -9.
\left(x^{2}-12x\right)+\left(3x-36\right)
Rewrite x^{2}-9x-36 as \left(x^{2}-12x\right)+\left(3x-36\right).
x\left(x-12\right)+3\left(x-12\right)
Factor out x in the first and 3 in the second group.
\left(x-12\right)\left(x+3\right)
Factor out common term x-12 by using distributive property.
x^{2}-9x-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(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-36\right)}}{2}
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(-9\right)±\sqrt{81-4\left(-36\right)}}{2}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81+144}}{2}
Multiply -4 times -36.
x=\frac{-\left(-9\right)±\sqrt{225}}{2}
Add 81 to 144.
x=\frac{-\left(-9\right)±15}{2}
Take the square root of 225.
x=\frac{9±15}{2}
The opposite of -9 is 9.
x=\frac{24}{2}
Now solve the equation x=\frac{9±15}{2} when ± is plus. Add 9 to 15.
x=12
Divide 24 by 2.
x=-\frac{6}{2}
Now solve the equation x=\frac{9±15}{2} when ± is minus. Subtract 15 from 9.
x=-3
Divide -6 by 2.
x^{2}-9x-36=\left(x-12\right)\left(x-\left(-3\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12 for x_{1} and -3 for x_{2}.
x^{2}-9x-36=\left(x-12\right)\left(x+3\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -9x -36 = 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 = 9 rs = -36
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{9}{2} - u s = \frac{9}{2} + u
Two numbers r and s sum up to 9 exactly when the average of the two numbers is \frac{1}{2}*9 = \frac{9}{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{9}{2} - u) (\frac{9}{2} + u) = -36
To solve for unknown quantity u, substitute these in the product equation rs = -36
\frac{81}{4} - u^2 = -36
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -36-\frac{81}{4} = -\frac{225}{4}
Simplify the expression by subtracting \frac{81}{4} on both sides
u^2 = \frac{225}{4} u = \pm\sqrt{\frac{225}{4}} = \pm \frac{15}{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{9}{2} - \frac{15}{2} = -3 s = \frac{9}{2} + \frac{15}{2} = 12
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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