Solve -x^2+200x-9900

$$- { x }^{ 2 } +200x-9900$$

Factor

$$\left(90-x\right)\left(x-110\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-9900. To find a and b, set up a system to be solved.

$$a+b=200 ab=-\left(-9900\right)=9900$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 9900.

$$1,9900 2,4950 3,3300 4,2475 5,1980 6,1650 9,1100 10,990 11,900 12,825 15,660 18,550 20,495 22,450 25,396 30,330 33,300 36,275 44,225 45,220 50,198 55,180 60,165 66,150 75,132 90,110 99,100$$

Calculate the sum for each pair.

$$1+9900=9901 2+4950=4952 3+3300=3303 4+2475=2479 5+1980=1985 6+1650=1656 9+1100=1109 10+990=1000 11+900=911 12+825=837 15+660=675 18+550=568 20+495=515 22+450=472 25+396=421 30+330=360 33+300=333 36+275=311 44+225=269 45+220=265 50+198=248 55+180=235 60+165=225 66+150=216 75+132=207 90+110=200 99+100=199$$

The solution is the pair that gives sum 200.

$$a=110 b=90$$

Rewrite -x^{2}+200x-9900 as \left(-x^{2}+110x\right)+\left(90x-9900\right).

$$\left(-x^{2}+110x\right)+\left(90x-9900\right)$$

Factor out -x in the first and 90 in the second group.

$$-x\left(x-110\right)+90\left(x-110\right)$$

Factor out common term x-110 by using distributive property.

$$\left(x-110\right)\left(-x+90\right)$$

Steps Using the Quadratic Formula

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.

$$-x^{2}+200x-9900=0$$

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.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-1\right)\left(-9900\right)}}{2\left(-1\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-1\right)\left(-9900\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-200±\sqrt{40000+4\left(-9900\right)}}{2\left(-1\right)}$$

Multiply 4 times -9900.

$$x=\frac{-200±\sqrt{40000-39600}}{2\left(-1\right)}$$

Add 40000 to -39600.

$$x=\frac{-200±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$x=\frac{-200±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-200±20}{-2}$$

Now solve the equation x=\frac{-200±20}{-2} when ± is plus. Add -200 to 20.

$$x=-\frac{180}{-2}$$

Divide -180 by -2.

$$x=90$$

Now solve the equation x=\frac{-200±20}{-2} when ± is minus. Subtract 20 from -200.

$$x=-\frac{220}{-2}$$

Divide -220 by -2.

$$x=110$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 90 for x_{1} and 110 for x_{2}.

$$-x^{2}+200x-9900=-\left(x-90\right)\left(x-110\right)$$

Evaluate

$$\left(90-x\right)\left(x-110\right)$$