Factor
\left(2-x\right)\left(7x+2\right)
Evaluate
\left(2-x\right)\left(7x+2\right)
Graph
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a+b=12 ab=-7\times 4=-28
Factor the expression by grouping. First, the expression needs to be rewritten as -7x^{2}+ax+bx+4. To find a and b, set up a system to be solved.
-1,28 -2,14 -4,7
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -28.
-1+28=27 -2+14=12 -4+7=3
Calculate the sum for each pair.
a=14 b=-2
The solution is the pair that gives sum 12.
\left(-7x^{2}+14x\right)+\left(-2x+4\right)
Rewrite -7x^{2}+12x+4 as \left(-7x^{2}+14x\right)+\left(-2x+4\right).
7x\left(-x+2\right)+2\left(-x+2\right)
Factor out 7x in the first and 2 in the second group.
\left(-x+2\right)\left(7x+2\right)
Factor out common term -x+2 by using distributive property.
-7x^{2}+12x+4=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{-12±\sqrt{12^{2}-4\left(-7\right)\times 4}}{2\left(-7\right)}
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{-12±\sqrt{144-4\left(-7\right)\times 4}}{2\left(-7\right)}
Square 12.
x=\frac{-12±\sqrt{144+28\times 4}}{2\left(-7\right)}
Multiply -4 times -7.
x=\frac{-12±\sqrt{144+112}}{2\left(-7\right)}
Multiply 28 times 4.
x=\frac{-12±\sqrt{256}}{2\left(-7\right)}
Add 144 to 112.
x=\frac{-12±16}{2\left(-7\right)}
Take the square root of 256.
x=\frac{-12±16}{-14}
Multiply 2 times -7.
x=\frac{4}{-14}
Now solve the equation x=\frac{-12±16}{-14} when ± is plus. Add -12 to 16.
x=-\frac{2}{7}
Reduce the fraction \frac{4}{-14} to lowest terms by extracting and canceling out 2.
x=-\frac{28}{-14}
Now solve the equation x=\frac{-12±16}{-14} when ± is minus. Subtract 16 from -12.
x=2
Divide -28 by -14.
-7x^{2}+12x+4=-7\left(x-\left(-\frac{2}{7}\right)\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 -\frac{2}{7} for x_{1} and 2 for x_{2}.
-7x^{2}+12x+4=-7\left(x+\frac{2}{7}\right)\left(x-2\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
-7x^{2}+12x+4=-7\times \frac{-7x-2}{-7}\left(x-2\right)
Add \frac{2}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
-7x^{2}+12x+4=\left(-7x-2\right)\left(x-2\right)
Cancel out 7, the greatest common factor in -7 and 7.
Examples
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Matrix
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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
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