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2\left(x^{2}-14x+24\right)
Factor out 2.
a+b=-14 ab=1\times 24=24
Consider x^{2}-14x+24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+24. To find a and b, set up a system to be solved.
-1,-24 -2,-12 -3,-8 -4,-6
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 24.
-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10
Calculate the sum for each pair.
a=-12 b=-2
The solution is the pair that gives sum -14.
\left(x^{2}-12x\right)+\left(-2x+24\right)
Rewrite x^{2}-14x+24 as \left(x^{2}-12x\right)+\left(-2x+24\right).
x\left(x-12\right)-2\left(x-12\right)
Factor out x in the first and -2 in the second group.
\left(x-12\right)\left(x-2\right)
Factor out common term x-12 by using distributive property.
2\left(x-12\right)\left(x-2\right)
Rewrite the complete factored expression.
2x^{2}-28x+48=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(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 48}}{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{-\left(-28\right)±\sqrt{784-4\times 2\times 48}}{2\times 2}
Square -28.
x=\frac{-\left(-28\right)±\sqrt{784-8\times 48}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-28\right)±\sqrt{784-384}}{2\times 2}
Multiply -8 times 48.
x=\frac{-\left(-28\right)±\sqrt{400}}{2\times 2}
Add 784 to -384.
x=\frac{-\left(-28\right)±20}{2\times 2}
Take the square root of 400.
x=\frac{28±20}{2\times 2}
The opposite of -28 is 28.
x=\frac{28±20}{4}
Multiply 2 times 2.
x=\frac{48}{4}
Now solve the equation x=\frac{28±20}{4} when ± is plus. Add 28 to 20.
x=12
Divide 48 by 4.
x=\frac{8}{4}
Now solve the equation x=\frac{28±20}{4} when ± is minus. Subtract 20 from 28.
x=2
Divide 8 by 4.
2x^{2}-28x+48=2\left(x-12\right)\left(x-2\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 2 for x_{2}.
x ^ 2 -14x +24 = 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 = 14 rs = 24
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 = 7 - u s = 7 + u
Two numbers r and s sum up to 14 exactly when the average of the two numbers is \frac{1}{2}*14 = 7. 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>
(7 - u) (7 + u) = 24
To solve for unknown quantity u, substitute these in the product equation rs = 24
49 - u^2 = 24
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 24-49 = -25
Simplify the expression by subtracting 49 on both sides
u^2 = 25 u = \pm\sqrt{25} = \pm 5
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =7 - 5 = 2 s = 7 + 5 = 12
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.