Solve -x^2+200x-9600

$$- { x }^{ 2 } +200x-9600$$

Factor

$$\left(80-x\right)\left(x-120\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-9600. To find a and b, set up a system to be solved.

$$a+b=200 ab=-\left(-9600\right)=9600$$

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 9600.

$$1,9600 2,4800 3,3200 4,2400 5,1920 6,1600 8,1200 10,960 12,800 15,640 16,600 20,480 24,400 25,384 30,320 32,300 40,240 48,200 50,192 60,160 64,150 75,128 80,120 96,100$$

Calculate the sum for each pair.

$$1+9600=9601 2+4800=4802 3+3200=3203 4+2400=2404 5+1920=1925 6+1600=1606 8+1200=1208 10+960=970 12+800=812 15+640=655 16+600=616 20+480=500 24+400=424 25+384=409 30+320=350 32+300=332 40+240=280 48+200=248 50+192=242 60+160=220 64+150=214 75+128=203 80+120=200 96+100=196$$

The solution is the pair that gives sum 200.

$$a=120 b=80$$

Rewrite -x^{2}+200x-9600 as \left(-x^{2}+120x\right)+\left(80x-9600\right).

$$\left(-x^{2}+120x\right)+\left(80x-9600\right)$$

Factor out -x in the first and 80 in the second group.

$$-x\left(x-120\right)+80\left(x-120\right)$$

Factor out common term x-120 by using distributive property.

$$\left(x-120\right)\left(-x+80\right)$$

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.

$$-x^{2}+200x-9600=0$$

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.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-1\right)\left(-9600\right)}}{2\left(-1\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-1\right)\left(-9600\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-200±\sqrt{40000+4\left(-9600\right)}}{2\left(-1\right)}$$

Multiply 4 times -9600.

$$x=\frac{-200±\sqrt{40000-38400}}{2\left(-1\right)}$$

Add 40000 to -38400.

$$x=\frac{-200±\sqrt{1600}}{2\left(-1\right)}$$

Take the square root of 1600.

$$x=\frac{-200±40}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-200±40}{-2}$$

Now solve the equation x=\frac{-200±40}{-2} when ± is plus. Add -200 to 40.

$$x=-\frac{160}{-2}$$

Divide -160 by -2.

$$x=80$$

Now solve the equation x=\frac{-200±40}{-2} when ± is minus. Subtract 40 from -200.

$$x=-\frac{240}{-2}$$

Divide -240 by -2.

$$x=120$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 80 for x_{1} and 120 for x_{2}.

$$-x^{2}+200x-9600=-\left(x-80\right)\left(x-120\right)$$

Evaluate

$$\left(80-x\right)\left(x-120\right)$$