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Factor the expression by grouping. First, the expression needs to be rewritten as . To find and , set up a system to be solved.
Since is negative, and have the opposite signs. Since is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product .
Calculate the sum for each pair.
The solution is the pair that gives sum .
Rewrite as .
Factor out in the first and in the second group.
Factor out common term by using distributive property.
Evaluate
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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.
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