Factor
\left(x-9\right)\left(5x-3\right)
Evaluate
\left(x-9\right)\left(5x-3\right)
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a+b=-48 ab=5\times 27=135
Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx+27. To find a and b, set up a system to be solved.
-1,-135 -3,-45 -5,-27 -9,-15
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 135.
-1-135=-136 -3-45=-48 -5-27=-32 -9-15=-24
Calculate the sum for each pair.
a=-45 b=-3
The solution is the pair that gives sum -48.
\left(5x^{2}-45x\right)+\left(-3x+27\right)
Rewrite 5x^{2}-48x+27 as \left(5x^{2}-45x\right)+\left(-3x+27\right).
5x\left(x-9\right)-3\left(x-9\right)
Factor out 5x in the first and -3 in the second group.
\left(x-9\right)\left(5x-3\right)
Factor out common term x-9 by using distributive property.
5x^{2}-48x+27=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(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 5\times 27}}{2\times 5}
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(-48\right)±\sqrt{2304-4\times 5\times 27}}{2\times 5}
Square -48.
x=\frac{-\left(-48\right)±\sqrt{2304-20\times 27}}{2\times 5}
Multiply -4 times 5.
x=\frac{-\left(-48\right)±\sqrt{2304-540}}{2\times 5}
Multiply -20 times 27.
x=\frac{-\left(-48\right)±\sqrt{1764}}{2\times 5}
Add 2304 to -540.
x=\frac{-\left(-48\right)±42}{2\times 5}
Take the square root of 1764.
x=\frac{48±42}{2\times 5}
The opposite of -48 is 48.
x=\frac{48±42}{10}
Multiply 2 times 5.
x=\frac{90}{10}
Now solve the equation x=\frac{48±42}{10} when ± is plus. Add 48 to 42.
x=9
Divide 90 by 10.
x=\frac{6}{10}
Now solve the equation x=\frac{48±42}{10} when ± is minus. Subtract 42 from 48.
x=\frac{3}{5}
Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.
5x^{2}-48x+27=5\left(x-9\right)\left(x-\frac{3}{5}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and \frac{3}{5} for x_{2}.
5x^{2}-48x+27=5\left(x-9\right)\times \frac{5x-3}{5}
Subtract \frac{3}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
5x^{2}-48x+27=\left(x-9\right)\left(5x-3\right)
Cancel out 5, the greatest common factor in 5 and 5.
Examples
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{ 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}