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$\exponential{(x)}{2} - 15 x + 50 $
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a+b=-15 ab=1\times 50=50
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+50. To find a and b, set up a system to be solved.
-1,-50 -2,-25 -5,-10
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 50.
-1-50=-51 -2-25=-27 -5-10=-15
Calculate the sum for each pair.
a=-10 b=-5
The solution is the pair that gives sum -15.
\left(x^{2}-10x\right)+\left(-5x+50\right)
Rewrite x^{2}-15x+50 as \left(x^{2}-10x\right)+\left(-5x+50\right).
x\left(x-10\right)-5\left(x-10\right)
Factor out x in the first and -5 in the second group.
\left(x-10\right)\left(x-5\right)
Factor out common term x-10 by using distributive property.
x^{2}-15x+50=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(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 50}}{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(-15\right)±\sqrt{225-4\times 50}}{2}
Square -15.
x=\frac{-\left(-15\right)±\sqrt{225-200}}{2}
Multiply -4 times 50.
x=\frac{-\left(-15\right)±\sqrt{25}}{2}
Add 225 to -200.
x=\frac{-\left(-15\right)±5}{2}
Take the square root of 25.
x=\frac{15±5}{2}
The opposite of -15 is 15.
x=\frac{20}{2}
Now solve the equation x=\frac{15±5}{2} when ± is plus. Add 15 to 5.
x=10
Divide 20 by 2.
x=\frac{10}{2}
Now solve the equation x=\frac{15±5}{2} when ± is minus. Subtract 5 from 15.
x=5
Divide 10 by 2.
x^{2}-15x+50=\left(x-10\right)\left(x-5\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and 5 for x_{2}.