Factor

\left(x-6\right)\left(x+2\right)

Evaluate

\left(x-6\right)\left(x+2\right)

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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.