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-x^{2}+27x+28
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=27 ab=-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 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=28 b=-1
The solution is the pair that gives sum 27.
\left(-x^{2}+28x\right)+\left(-x+28\right)
Rewrite -x^{2}+27x+28 as \left(-x^{2}+28x\right)+\left(-x+28\right).
-x\left(x-28\right)-\left(x-28\right)
Factor out -x in the first and -1 in the second group.
\left(x-28\right)\left(-x-1\right)
Factor out common term x-28 by using distributive property.
-x^{2}+27x+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{-27±\sqrt{27^{2}-4\left(-1\right)\times 28}}{2\left(-1\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{-27±\sqrt{729-4\left(-1\right)\times 28}}{2\left(-1\right)}
Square 27.
x=\frac{-27±\sqrt{729+4\times 28}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-27±\sqrt{729+112}}{2\left(-1\right)}
Multiply 4 times 28.
x=\frac{-27±\sqrt{841}}{2\left(-1\right)}
Add 729 to 112.
x=\frac{-27±29}{2\left(-1\right)}
Take the square root of 841.
x=\frac{-27±29}{-2}
Multiply 2 times -1.
x=\frac{2}{-2}
Now solve the equation x=\frac{-27±29}{-2} when ± is plus. Add -27 to 29.
x=-1
Divide 2 by -2.
x=-\frac{56}{-2}
Now solve the equation x=\frac{-27±29}{-2} when ± is minus. Subtract 29 from -27.
x=28
Divide -56 by -2.
-x^{2}+27x+28=-\left(x-\left(-1\right)\right)\left(x-28\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and 28 for x_{2}.
-x^{2}+27x+28=-\left(x+1\right)\left(x-28\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.