Factor
\left(x-14\right)\left(x-2\right)
Evaluate
\left(x-14\right)\left(x-2\right)
Graph
Share
Copied to clipboard
a+b=-16 ab=1\times 28=28
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.
-1,-28 -2,-14 -4,-7
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 28.
-1-28=-29 -2-14=-16 -4-7=-11
Calculate the sum for each pair.
a=-14 b=-2
The solution is the pair that gives sum -16.
\left(x^{2}-14x\right)+\left(-2x+28\right)
Rewrite x^{2}-16x+28 as \left(x^{2}-14x\right)+\left(-2x+28\right).
x\left(x-14\right)-2\left(x-14\right)
Factor out x in the first and -2 in the second group.
\left(x-14\right)\left(x-2\right)
Factor out common term x-14 by using distributive property.
x^{2}-16x+28=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(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 28}}{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(-16\right)±\sqrt{256-4\times 28}}{2}
Square -16.
x=\frac{-\left(-16\right)±\sqrt{256-112}}{2}
Multiply -4 times 28.
x=\frac{-\left(-16\right)±\sqrt{144}}{2}
Add 256 to -112.
x=\frac{-\left(-16\right)±12}{2}
Take the square root of 144.
x=\frac{16±12}{2}
The opposite of -16 is 16.
x=\frac{28}{2}
Now solve the equation x=\frac{16±12}{2} when ± is plus. Add 16 to 12.
x=14
Divide 28 by 2.
x=\frac{4}{2}
Now solve the equation x=\frac{16±12}{2} when ± is minus. Subtract 12 from 16.
x=2
Divide 4 by 2.
x^{2}-16x+28=\left(x-14\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 14 for x_{1} and 2 for x_{2}.
x ^ 2 -16x +28 = 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 = 16 rs = 28
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 = 8 - u s = 8 + u
Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>
(8 - u) (8 + u) = 28
To solve for unknown quantity u, substitute these in the product equation rs = 28
64 - u^2 = 28
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 28-64 = -36
Simplify the expression by subtracting 64 on both sides
u^2 = 36 u = \pm\sqrt{36} = \pm 6
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =8 - 6 = 2 s = 8 + 6 = 14
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
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}