Solve -2x^2+380x-6800

$$-2 { x }^{ 2 } +380x-6800$$

Factor

$$2\left(20-x\right)\left(x-170\right)$$

Solution Steps

Factor out 2.

$$2\left(-x^{2}+190x-3400\right)$$

Consider -x^{2}+190x-3400. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-3400. To find a and b, set up a system to be solved.

$$a+b=190 ab=-\left(-3400\right)=3400$$

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 3400.

$$1,3400 2,1700 4,850 5,680 8,425 10,340 17,200 20,170 25,136 34,100 40,85 50,68$$

Calculate the sum for each pair.

$$1+3400=3401 2+1700=1702 4+850=854 5+680=685 8+425=433 10+340=350 17+200=217 20+170=190 25+136=161 34+100=134 40+85=125 50+68=118$$

The solution is the pair that gives sum 190.

$$a=170 b=20$$

Rewrite -x^{2}+190x-3400 as \left(-x^{2}+170x\right)+\left(20x-3400\right).

$$\left(-x^{2}+170x\right)+\left(20x-3400\right)$$

Factor out -x in the first and 20 in the second group.

$$-x\left(x-170\right)+20\left(x-170\right)$$

Factor out common term x-170 by using distributive property.

$$\left(x-170\right)\left(-x+20\right)$$

Rewrite the complete factored expression.

$$2\left(x-170\right)\left(-x+20\right)$$

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.

$$-2x^{2}+380x-6800=0$$

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.

$$x=\frac{-380±\sqrt{380^{2}-4\left(-2\right)\left(-6800\right)}}{2\left(-2\right)}$$

Square 380.

$$x=\frac{-380±\sqrt{144400-4\left(-2\right)\left(-6800\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-380±\sqrt{144400+8\left(-6800\right)}}{2\left(-2\right)}$$

Multiply 8 times -6800.

$$x=\frac{-380±\sqrt{144400-54400}}{2\left(-2\right)}$$

Add 144400 to -54400.

$$x=\frac{-380±\sqrt{90000}}{2\left(-2\right)}$$

Take the square root of 90000.

$$x=\frac{-380±300}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-380±300}{-4}$$

Now solve the equation x=\frac{-380±300}{-4} when ± is plus. Add -380 to 300.

$$x=-\frac{80}{-4}$$

Divide -80 by -4.

$$x=20$$

Now solve the equation x=\frac{-380±300}{-4} when ± is minus. Subtract 300 from -380.

$$x=-\frac{680}{-4}$$

Divide -680 by -4.

$$x=170$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 20 for x_{1} and 170 for x_{2}.

$$-2x^{2}+380x-6800=-2\left(x-20\right)\left(x-170\right)$$

Evaluate

$$2\left(20-x\right)\left(x-170\right)$$