Type a math problem  Factor  Solution Steps
Steps Using Direct Factoring Method
Solution Steps
Factor out .
Consider . 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 positive, and have the same sign. Since is positive, and are both 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.
Rewrite the complete factored expression.
Evaluate Graph Giving is as easy as 1, 2, 3
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2\left(x^{2}+8x+12\right)
Factor out 2.
a+b=8 ab=1\times 12=12
Consider x^{2}+8x+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 positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.
1+12=13 2+6=8 3+4=7
Calculate the sum for each pair.
a=2 b=6
The solution is the pair that gives sum 8.
\left(x^{2}+2x\right)+\left(6x+12\right)
Rewrite x^{2}+8x+12 as \left(x^{2}+2x\right)+\left(6x+12\right).
x\left(x+2\right)+6\left(x+2\right)
Factor out x in the first and 6 in the second group.
\left(x+2\right)\left(x+6\right)
Factor out common term x+2 by using distributive property.
2\left(x+2\right)\left(x+6\right)
Rewrite the complete factored expression.
2x^{2}+16x+24=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{-16±\sqrt{16^{2}-4\times 2\times 24}}{2\times 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{-16±\sqrt{256-4\times 2\times 24}}{2\times 2}
Square 16.
x=\frac{-16±\sqrt{256-8\times 24}}{2\times 2}
Multiply -4 times 2.
x=\frac{-16±\sqrt{256-192}}{2\times 2}
Multiply -8 times 24.
x=\frac{-16±\sqrt{64}}{2\times 2}
x=\frac{-16±8}{2\times 2}
Take the square root of 64.
x=\frac{-16±8}{4}
Multiply 2 times 2.
x=\frac{-8}{4}
Now solve the equation x=\frac{-16±8}{4} when ± is plus. Add -16 to 8.
x=-2
Divide -8 by 4.
x=\frac{-24}{4}
Now solve the equation x=\frac{-16±8}{4} when ± is minus. Subtract 8 from -16.
x=-6
Divide -24 by 4.
2x^{2}+16x+24=2\left(x-\left(-2\right)\right)\left(x-\left(-6\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and -6 for x_{2}.
2x^{2}+16x+24=2\left(x+2\right)\left(x+6\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +8x +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.This is achieved by dividing both sides of the equation by 2
r + s = -8 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 = -4 - u s = -4 + u
Two numbers r and s sum up to -8 exactly when the average of the two numbers is \frac{1}{2}*-8 = -4. 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>
(-4 - u) (-4 + u) = 12
To solve for unknown quantity u, substitute these in the product equation rs = 12
16 - u^2 = 12
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 12-16 = -4
Simplify the expression by subtracting 16 on both sides
u^2 = 4 u = \pm\sqrt{4} = \pm 2
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-4 - 2 = -6 s = -4 + 2 = -2
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.