Factor
2\left(x+4\right)\left(x+6\right)
Evaluate
2\left(x+4\right)\left(x+6\right)
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2\left(x^{2}+10x+24\right)
Factor out 2.
a+b=10 ab=1\times 24=24
Consider x^{2}+10x+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 positive, a and b are both positive. 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=4 b=6
The solution is the pair that gives sum 10.
\left(x^{2}+4x\right)+\left(6x+24\right)
Rewrite x^{2}+10x+24 as \left(x^{2}+4x\right)+\left(6x+24\right).
x\left(x+4\right)+6\left(x+4\right)
Factor out x in the first and 6 in the second group.
\left(x+4\right)\left(x+6\right)
Factor out common term x+4 by using distributive property.
2\left(x+4\right)\left(x+6\right)
Rewrite the complete factored expression.
2x^{2}+20x+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{-20±\sqrt{20^{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{-20±\sqrt{400-4\times 2\times 48}}{2\times 2}
Square 20.
x=\frac{-20±\sqrt{400-8\times 48}}{2\times 2}
Multiply -4 times 2.
x=\frac{-20±\sqrt{400-384}}{2\times 2}
Multiply -8 times 48.
x=\frac{-20±\sqrt{16}}{2\times 2}
Add 400 to -384.
x=\frac{-20±4}{2\times 2}
Take the square root of 16.
x=\frac{-20±4}{4}
Multiply 2 times 2.
x=-\frac{16}{4}
Now solve the equation x=\frac{-20±4}{4} when ± is plus. Add -20 to 4.
x=-4
Divide -16 by 4.
x=-\frac{24}{4}
Now solve the equation x=\frac{-20±4}{4} when ± is minus. Subtract 4 from -20.
x=-6
Divide -24 by 4.
2x^{2}+20x+48=2\left(x-\left(-4\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 -4 for x_{1} and -6 for x_{2}.
2x^{2}+20x+48=2\left(x+4\right)\left(x+6\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 +10x +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 = -10 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 = -5 - u s = -5 + u
Two numbers r and s sum up to -10 exactly when the average of the two numbers is \frac{1}{2}*-10 = -5. 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>
(-5 - u) (-5 + u) = 24
To solve for unknown quantity u, substitute these in the product equation rs = 24
25 - u^2 = 24
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 24-25 = -1
Simplify the expression by subtracting 25 on both sides
u^2 = 1 u = \pm\sqrt{1} = \pm 1
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-5 - 1 = -6 s = -5 + 1 = -4
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}