Solve -10x^2+1000x-21000

$$-10 { x }^{ 2 } +1000x-21000$$

Factor

$$10\left(30-x\right)\left(x-70\right)$$

Solution Steps

Factor out 10.

$$10\left(-x^{2}+100x-2100\right)$$

Consider -x^{2}+100x-2100. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-2100. To find a and b, set up a system to be solved.

$$a+b=100 ab=-\left(-2100\right)=2100$$

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 2100.

$$1,2100 2,1050 3,700 4,525 5,420 6,350 7,300 10,210 12,175 14,150 15,140 20,105 21,100 25,84 28,75 30,70 35,60 42,50$$

Calculate the sum for each pair.

$$1+2100=2101 2+1050=1052 3+700=703 4+525=529 5+420=425 6+350=356 7+300=307 10+210=220 12+175=187 14+150=164 15+140=155 20+105=125 21+100=121 25+84=109 28+75=103 30+70=100 35+60=95 42+50=92$$

The solution is the pair that gives sum 100.

$$a=70 b=30$$

Rewrite -x^{2}+100x-2100 as \left(-x^{2}+70x\right)+\left(30x-2100\right).

$$\left(-x^{2}+70x\right)+\left(30x-2100\right)$$

Factor out -x in the first and 30 in the second group.

$$-x\left(x-70\right)+30\left(x-70\right)$$

Factor out common term x-70 by using distributive property.

$$\left(x-70\right)\left(-x+30\right)$$

Rewrite the complete factored expression.

$$10\left(x-70\right)\left(-x+30\right)$$

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.

$$-10x^{2}+1000x-21000=0$$

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.

$$x=\frac{-1000±\sqrt{1000^{2}-4\left(-10\right)\left(-21000\right)}}{2\left(-10\right)}$$

Square 1000.

$$x=\frac{-1000±\sqrt{1000000-4\left(-10\right)\left(-21000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1000±\sqrt{1000000+40\left(-21000\right)}}{2\left(-10\right)}$$

Multiply 40 times -21000.

$$x=\frac{-1000±\sqrt{1000000-840000}}{2\left(-10\right)}$$

Add 1000000 to -840000.

$$x=\frac{-1000±\sqrt{160000}}{2\left(-10\right)}$$

Take the square root of 160000.

$$x=\frac{-1000±400}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1000±400}{-20}$$

Now solve the equation x=\frac{-1000±400}{-20} when ± is plus. Add -1000 to 400.

$$x=-\frac{600}{-20}$$

Divide -600 by -20.

$$x=30$$

Now solve the equation x=\frac{-1000±400}{-20} when ± is minus. Subtract 400 from -1000.

$$x=-\frac{1400}{-20}$$

Divide -1400 by -20.

$$x=70$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 30 for x_{1} and 70 for x_{2}.

$$-10x^{2}+1000x-21000=-10\left(x-30\right)\left(x-70\right)$$

Evaluate

$$10\left(30-x\right)\left(x-70\right)$$