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a+b=-4 ab=1\left(-12\right)=-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=-6 b=2
The solution is the pair that gives sum -4.
\left(x^{2}-6x\right)+\left(2x-12\right)
Rewrite x^{2}-4x-12 as \left(x^{2}-6x\right)+\left(2x-12\right).
x\left(x-6\right)+2\left(x-6\right)
Factor out x in the first and 2 in the second group.
\left(x-6\right)\left(x+2\right)
Factor out common term x-6 by using distributive property.
x^{2}-4x-12=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(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-12\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(-4\right)±\sqrt{16-4\left(-12\right)}}{2}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16+48}}{2}
Multiply -4 times -12.
x=\frac{-\left(-4\right)±\sqrt{64}}{2}
Add 16 to 48.
x=\frac{-\left(-4\right)±8}{2}
Take the square root of 64.
x=\frac{4±8}{2}
The opposite of -4 is 4.
x=\frac{12}{2}
Now solve the equation x=\frac{4±8}{2} when ± is plus. Add 4 to 8.
x=6
Divide 12 by 2.
x=\frac{-4}{2}
Now solve the equation x=\frac{4±8}{2} when ± is minus. Subtract 8 from 4.
x=-2
Divide -4 by 2.
x^{2}-4x-12=\left(x-6\right)\left(x-\left(-2\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 -2 for x_{2}.
x^{2}-4x-12=\left(x-6\right)\left(x+2\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -4x -12 = 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 = 4 rs = -12
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 = 2 - u s = 2 + u
Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 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>
(2 - u) (2 + u) = -12
To solve for unknown quantity u, substitute these in the product equation rs = -12
4 - u^2 = -12
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -12-4 = -16
Simplify the expression by subtracting 4 on both sides
u^2 = 16 u = \pm\sqrt{16} = \pm 4
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =2 - 4 = -2 s = 2 + 4 = 6
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.