Solve 350/v-350/v+20=2

$$\frac { 350 } { v } - \frac { 350 } { v + 20 } = 2$$

Solve for v

$$v=-70$$

$$v=50$$

Steps Using Factoring By Grouping

Variable v cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by v\left(v+20\right), the least common multiple of v,v+20.

$$\left(v+20\right)\times 350-v\times 350=2v\left(v+20\right)$$

Use the distributive property to multiply v+20 by 350.

$$350v+7000-v\times 350=2v\left(v+20\right)$$

Use the distributive property to multiply 2v by v+20.

$$350v+7000-v\times 350=2v^{2}+40v$$

Subtract 2v^{2} from both sides.

$$350v+7000-v\times 350-2v^{2}=40v$$

Subtract 40v from both sides.

$$350v+7000-v\times 350-2v^{2}-40v=0$$

Combine 350v and -40v to get 310v.

$$310v+7000-v\times 350-2v^{2}=0$$

Multiply -1 and 350 to get -350.

$$310v+7000-350v-2v^{2}=0$$

Combine 310v and -350v to get -40v.

$$-40v+7000-2v^{2}=0$$

Divide both sides by 2.

$$-20v+3500-v^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-v^{2}-20v+3500=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -v^{2}+av+bv+3500. To find a and b, set up a system to be solved.

$$a+b=-20 ab=-3500=-3500$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -3500.

$$1,-3500 2,-1750 4,-875 5,-700 7,-500 10,-350 14,-250 20,-175 25,-140 28,-125 35,-100 50,-70$$

Calculate the sum for each pair.

$$1-3500=-3499 2-1750=-1748 4-875=-871 5-700=-695 7-500=-493 10-350=-340 14-250=-236 20-175=-155 25-140=-115 28-125=-97 35-100=-65 50-70=-20$$

The solution is the pair that gives sum -20.

$$a=50 b=-70$$

Rewrite -v^{2}-20v+3500 as \left(-v^{2}+50v\right)+\left(-70v+3500\right).

$$\left(-v^{2}+50v\right)+\left(-70v+3500\right)$$

Factor out v in the first and 70 in the second group.

$$v\left(-v+50\right)+70\left(-v+50\right)$$

Factor out common term -v+50 by using distributive property.

$$\left(-v+50\right)\left(v+70\right)$$

To find equation solutions, solve -v+50=0 and v+70=0.

$$v=50 v=-70$$

Steps Using the Quadratic Formula

Variable v cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by v\left(v+20\right), the least common multiple of v,v+20.

$$\left(v+20\right)\times 350-v\times 350=2v\left(v+20\right)$$

Use the distributive property to multiply v+20 by 350.

$$350v+7000-v\times 350=2v\left(v+20\right)$$

Use the distributive property to multiply 2v by v+20.

$$350v+7000-v\times 350=2v^{2}+40v$$

Subtract 2v^{2} from both sides.

$$350v+7000-v\times 350-2v^{2}=40v$$

Subtract 40v from both sides.

$$350v+7000-v\times 350-2v^{2}-40v=0$$

Combine 350v and -40v to get 310v.

$$310v+7000-v\times 350-2v^{2}=0$$

Multiply -1 and 350 to get -350.

$$310v+7000-350v-2v^{2}=0$$

Combine 310v and -350v to get -40v.

$$-40v+7000-2v^{2}=0$$

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.

$$-2v^{2}-40v+7000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -40 for b, and 7000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\left(-2\right)\times 7000}}{2\left(-2\right)}$$

Square -40.

$$v=\frac{-\left(-40\right)±\sqrt{1600-4\left(-2\right)\times 7000}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$v=\frac{-\left(-40\right)±\sqrt{1600+8\times 7000}}{2\left(-2\right)}$$

Multiply 8 times 7000.

$$v=\frac{-\left(-40\right)±\sqrt{1600+56000}}{2\left(-2\right)}$$

Add 1600 to 56000.

$$v=\frac{-\left(-40\right)±\sqrt{57600}}{2\left(-2\right)}$$

Take the square root of 57600.

$$v=\frac{-\left(-40\right)±240}{2\left(-2\right)}$$

The opposite of -40 is 40.

$$v=\frac{40±240}{2\left(-2\right)}$$

Multiply 2 times -2.

$$v=\frac{40±240}{-4}$$

Now solve the equation v=\frac{40±240}{-4} when ± is plus. Add 40 to 240.

$$v=\frac{280}{-4}$$

Divide 280 by -4.

$$v=-70$$

Now solve the equation v=\frac{40±240}{-4} when ± is minus. Subtract 240 from 40.

$$v=-\frac{200}{-4}$$

Divide -200 by -4.

$$v=50$$

The equation is now solved.

$$v=-70 v=50$$

Steps for Completing the Square

Variable v cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by v\left(v+20\right), the least common multiple of v,v+20.

$$\left(v+20\right)\times 350-v\times 350=2v\left(v+20\right)$$

Use the distributive property to multiply v+20 by 350.

$$350v+7000-v\times 350=2v\left(v+20\right)$$

Use the distributive property to multiply 2v by v+20.

$$350v+7000-v\times 350=2v^{2}+40v$$

Subtract 2v^{2} from both sides.

$$350v+7000-v\times 350-2v^{2}=40v$$

Subtract 40v from both sides.

$$350v+7000-v\times 350-2v^{2}-40v=0$$

Combine 350v and -40v to get 310v.

$$310v+7000-v\times 350-2v^{2}=0$$

Subtract 7000 from both sides. Anything subtracted from zero gives its negation.

$$310v-v\times 350-2v^{2}=-7000$$

Multiply -1 and 350 to get -350.

$$310v-350v-2v^{2}=-7000$$

Combine 310v and -350v to get -40v.

$$-40v-2v^{2}=-7000$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2v^{2}-40v=-7000$$

Divide both sides by -2.

$$\frac{-2v^{2}-40v}{-2}=-\frac{7000}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$v^{2}+\left(-\frac{40}{-2}\right)v=-\frac{7000}{-2}$$

Divide -40 by -2.

$$v^{2}+20v=-\frac{7000}{-2}$$

Divide -7000 by -2.

$$v^{2}+20v=3500$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+20v+10^{2}=3500+10^{2}$$

Square 10.

$$v^{2}+20v+100=3500+100$$

Add 3500 to 100.

$$v^{2}+20v+100=3600$$

Factor v^{2}+20v+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+10\right)^{2}=3600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+10\right)^{2}}=\sqrt{3600}$$

Simplify.

$$v+10=60 v+10=-60$$

Subtract 10 from both sides of the equation.

$$v=50 v=-70$$