Solve 10^-3(v-10/2+v-8/4+v-10/5)*v/2-10^-3+v*10^3=0

$$10 ^ { - 3 } ( \frac { v - 10 } { 2 } + \frac { v - 8 } { 4 } + \frac { v - 10 } { 5 } ) \cdot \frac { v } { 2 } - 10 ^ { - 3 } + v \cdot 10 ^ { 3 } = 0$$

Solve for v

$$v=\frac{2\sqrt{99999100002215}-19999910}{19}\approx 0.000001$$

$$v=\frac{-2\sqrt{99999100002215}-19999910}{19}\approx -2105253.684211526$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 20, the least common multiple of 2,4,5.

$$20\left(10^{-3}\left(\frac{v-10}{2}+\frac{v-8}{4}+\frac{v-10}{5}\right)\times \frac{v}{2}-10^{-3}\right)+20v\times 10^{3}=0$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$20\left(\frac{1}{1000}\left(\frac{v-10}{2}+\frac{v-8}{4}+\frac{v-10}{5}\right)\times \frac{v}{2}-10^{-3}\right)+20v\times 10^{3}=0$$

Combine \frac{v-10}{2} and \frac{v-10}{5} to get \frac{7}{10}\left(v-10\right).

$$20\left(\frac{1}{1000}\left(\frac{7}{10}\left(v-10\right)+\frac{v-8}{4}\right)\times \frac{v}{2}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply \frac{1}{1000} times \frac{v}{2} by multiplying numerator times numerator and denominator times denominator.

$$20\left(\frac{v}{1000\times 2}\left(\frac{7}{10}\left(v-10\right)+\frac{v-8}{4}\right)-10^{-3}\right)+20v\times 10^{3}=0$$

Use the distributive property to multiply \frac{v}{1000\times 2} by \frac{7}{10}\left(v-10\right)+\frac{v-8}{4}.

$$20\left(\frac{7}{10}\times \frac{v}{1000\times 2}\left(v-10\right)+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply 1000 and 2 to get 2000.

$$20\left(\frac{7}{10}\times \frac{v}{2000}\left(v-10\right)+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply \frac{7}{10} times \frac{v}{2000} by multiplying numerator times numerator and denominator times denominator.

$$20\left(\frac{7v}{10\times 2000}\left(v-10\right)+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Express \frac{7v}{10\times 2000}\left(v-10\right) as a single fraction.

$$20\left(\frac{7v\left(v-10\right)}{10\times 2000}+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply 1000 and 2 to get 2000.

$$20\left(\frac{7v\left(v-10\right)}{10\times 2000}+\frac{v}{2000}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply \frac{v}{2000} times \frac{v-8}{4} by multiplying numerator times numerator and denominator times denominator.

$$20\left(\frac{7v\left(v-10\right)}{10\times 2000}+\frac{v\left(v-8\right)}{2000\times 4}-10^{-3}\right)+20v\times 10^{3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10\times 2000 and 2000\times 4 is 20\times 2000. Multiply \frac{7v\left(v-10\right)}{10\times 2000} times \frac{2}{2}. Multiply \frac{v\left(v-8\right)}{2000\times 4} times \frac{5}{5}.

$$20\left(\frac{2\times 7v\left(v-10\right)}{20\times 2000}+\frac{5v\left(v-8\right)}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Since \frac{2\times 7v\left(v-10\right)}{20\times 2000} and \frac{5v\left(v-8\right)}{20\times 2000} have the same denominator, add them by adding their numerators.

$$20\left(\frac{2\times 7v\left(v-10\right)+5v\left(v-8\right)}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Do the multiplications in 2\times 7v\left(v-10\right)+5v\left(v-8\right).

$$20\left(\frac{14v^{2}-140v+5v^{2}-40v}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Combine like terms in 14v^{2}-140v+5v^{2}-40v.

$$20\left(\frac{19v^{2}-180v}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply 20 and 2000 to get 40000.

$$20\left(\frac{19v^{2}-180v}{40000}-10^{-3}\right)+20v\times 10^{3}=0$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$20\left(\frac{19v^{2}-180v}{40000}-\frac{1}{1000}\right)+20v\times 10^{3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 40000 and 1000 is 40000. Multiply \frac{1}{1000} times \frac{40}{40}.

$$20\left(\frac{19v^{2}-180v}{40000}-\frac{40}{40000}\right)+20v\times 10^{3}=0$$

Since \frac{19v^{2}-180v}{40000} and \frac{40}{40000} have the same denominator, subtract them by subtracting their numerators.

$$20\times \frac{19v^{2}-180v-40}{40000}+20v\times 10^{3}=0$$

Cancel out 40000, the greatest common factor in 20 and 40000.

$$\frac{19v^{2}-180v-40}{2000}+20v\times 10^{3}=0$$

Calculate 10 to the power of 3 and get 1000.

$$\frac{19v^{2}-180v-40}{2000}+20v\times 1000=0$$

Multiply 20 and 1000 to get 20000.

$$\frac{19v^{2}-180v-40}{2000}+20000v=0$$

Divide each term of 19v^{2}-180v-40 by 2000 to get \frac{19}{2000}v^{2}-\frac{9}{100}v-\frac{1}{50}.

$$\frac{19}{2000}v^{2}-\frac{9}{100}v-\frac{1}{50}+20000v=0$$

Combine -\frac{9}{100}v and 20000v to get \frac{1999991}{100}v.

$$\frac{19}{2000}v^{2}+\frac{1999991}{100}v-\frac{1}{50}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{19}{2000} for a, \frac{1999991}{100} for b, and -\frac{1}{50} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\frac{1999991}{100}±\sqrt{\left(\frac{1999991}{100}\right)^{2}-4\times \frac{19}{2000}\left(-\frac{1}{50}\right)}}{2\times \frac{19}{2000}}$$

Square \frac{1999991}{100} by squaring both the numerator and the denominator of the fraction.

$$v=\frac{-\frac{1999991}{100}±\sqrt{\frac{3999964000081}{10000}-4\times \frac{19}{2000}\left(-\frac{1}{50}\right)}}{2\times \frac{19}{2000}}$$

Multiply -4 times \frac{19}{2000}.

$$v=\frac{-\frac{1999991}{100}±\sqrt{\frac{3999964000081}{10000}-\frac{19}{500}\left(-\frac{1}{50}\right)}}{2\times \frac{19}{2000}}$$

Multiply -\frac{19}{500} times -\frac{1}{50} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$v=\frac{-\frac{1999991}{100}±\sqrt{\frac{3999964000081}{10000}+\frac{19}{25000}}}{2\times \frac{19}{2000}}$$

Add \frac{3999964000081}{10000} to \frac{19}{25000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v=\frac{-\frac{1999991}{100}±\sqrt{\frac{19999820000443}{50000}}}{2\times \frac{19}{2000}}$$

Take the square root of \frac{19999820000443}{50000}.

$$v=\frac{-\frac{1999991}{100}±\frac{\sqrt{99999100002215}}{500}}{2\times \frac{19}{2000}}$$

Multiply 2 times \frac{19}{2000}.

$$v=\frac{-\frac{1999991}{100}±\frac{\sqrt{99999100002215}}{500}}{\frac{19}{1000}}$$

Now solve the equation v=\frac{-\frac{1999991}{100}±\frac{\sqrt{99999100002215}}{500}}{\frac{19}{1000}} when ± is plus. Add -\frac{1999991}{100} to \frac{\sqrt{99999100002215}}{500}.

$$v=\frac{\frac{\sqrt{99999100002215}}{500}-\frac{1999991}{100}}{\frac{19}{1000}}$$

Divide -\frac{1999991}{100}+\frac{\sqrt{99999100002215}}{500} by \frac{19}{1000} by multiplying -\frac{1999991}{100}+\frac{\sqrt{99999100002215}}{500} by the reciprocal of \frac{19}{1000}.

$$v=\frac{2\sqrt{99999100002215}-19999910}{19}$$

Now solve the equation v=\frac{-\frac{1999991}{100}±\frac{\sqrt{99999100002215}}{500}}{\frac{19}{1000}} when ± is minus. Subtract \frac{\sqrt{99999100002215}}{500} from -\frac{1999991}{100}.

$$v=\frac{-\frac{\sqrt{99999100002215}}{500}-\frac{1999991}{100}}{\frac{19}{1000}}$$

Divide -\frac{1999991}{100}-\frac{\sqrt{99999100002215}}{500} by \frac{19}{1000} by multiplying -\frac{1999991}{100}-\frac{\sqrt{99999100002215}}{500} by the reciprocal of \frac{19}{1000}.

$$v=\frac{-2\sqrt{99999100002215}-19999910}{19}$$

The equation is now solved.

$$v=\frac{2\sqrt{99999100002215}-19999910}{19} v=\frac{-2\sqrt{99999100002215}-19999910}{19}$$

Steps for Completing the Square

Multiply both sides of the equation by 20, the least common multiple of 2,4,5.

$$20\left(10^{-3}\left(\frac{v-10}{2}+\frac{v-8}{4}+\frac{v-10}{5}\right)\times \frac{v}{2}-10^{-3}\right)+20v\times 10^{3}=0$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$20\left(\frac{1}{1000}\left(\frac{v-10}{2}+\frac{v-8}{4}+\frac{v-10}{5}\right)\times \frac{v}{2}-10^{-3}\right)+20v\times 10^{3}=0$$

Combine \frac{v-10}{2} and \frac{v-10}{5} to get \frac{7}{10}\left(v-10\right).

$$20\left(\frac{1}{1000}\left(\frac{7}{10}\left(v-10\right)+\frac{v-8}{4}\right)\times \frac{v}{2}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply \frac{1}{1000} times \frac{v}{2} by multiplying numerator times numerator and denominator times denominator.

$$20\left(\frac{v}{1000\times 2}\left(\frac{7}{10}\left(v-10\right)+\frac{v-8}{4}\right)-10^{-3}\right)+20v\times 10^{3}=0$$

Use the distributive property to multiply \frac{v}{1000\times 2} by \frac{7}{10}\left(v-10\right)+\frac{v-8}{4}.

$$20\left(\frac{7}{10}\times \frac{v}{1000\times 2}\left(v-10\right)+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply 1000 and 2 to get 2000.

$$20\left(\frac{7}{10}\times \frac{v}{2000}\left(v-10\right)+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply \frac{7}{10} times \frac{v}{2000} by multiplying numerator times numerator and denominator times denominator.

$$20\left(\frac{7v}{10\times 2000}\left(v-10\right)+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Express \frac{7v}{10\times 2000}\left(v-10\right) as a single fraction.

$$20\left(\frac{7v\left(v-10\right)}{10\times 2000}+\frac{v}{1000\times 2}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply 1000 and 2 to get 2000.

$$20\left(\frac{7v\left(v-10\right)}{10\times 2000}+\frac{v}{2000}\times \frac{v-8}{4}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply \frac{v}{2000} times \frac{v-8}{4} by multiplying numerator times numerator and denominator times denominator.

$$20\left(\frac{7v\left(v-10\right)}{10\times 2000}+\frac{v\left(v-8\right)}{2000\times 4}-10^{-3}\right)+20v\times 10^{3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10\times 2000 and 2000\times 4 is 20\times 2000. Multiply \frac{7v\left(v-10\right)}{10\times 2000} times \frac{2}{2}. Multiply \frac{v\left(v-8\right)}{2000\times 4} times \frac{5}{5}.

$$20\left(\frac{2\times 7v\left(v-10\right)}{20\times 2000}+\frac{5v\left(v-8\right)}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Since \frac{2\times 7v\left(v-10\right)}{20\times 2000} and \frac{5v\left(v-8\right)}{20\times 2000} have the same denominator, add them by adding their numerators.

$$20\left(\frac{2\times 7v\left(v-10\right)+5v\left(v-8\right)}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Do the multiplications in 2\times 7v\left(v-10\right)+5v\left(v-8\right).

$$20\left(\frac{14v^{2}-140v+5v^{2}-40v}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Combine like terms in 14v^{2}-140v+5v^{2}-40v.

$$20\left(\frac{19v^{2}-180v}{20\times 2000}-10^{-3}\right)+20v\times 10^{3}=0$$

Multiply 20 and 2000 to get 40000.

$$20\left(\frac{19v^{2}-180v}{40000}-10^{-3}\right)+20v\times 10^{3}=0$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$20\left(\frac{19v^{2}-180v}{40000}-\frac{1}{1000}\right)+20v\times 10^{3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 40000 and 1000 is 40000. Multiply \frac{1}{1000} times \frac{40}{40}.

$$20\left(\frac{19v^{2}-180v}{40000}-\frac{40}{40000}\right)+20v\times 10^{3}=0$$

Since \frac{19v^{2}-180v}{40000} and \frac{40}{40000} have the same denominator, subtract them by subtracting their numerators.

$$20\times \frac{19v^{2}-180v-40}{40000}+20v\times 10^{3}=0$$

Cancel out 40000, the greatest common factor in 20 and 40000.

$$\frac{19v^{2}-180v-40}{2000}+20v\times 10^{3}=0$$

Calculate 10 to the power of 3 and get 1000.

$$\frac{19v^{2}-180v-40}{2000}+20v\times 1000=0$$

Multiply 20 and 1000 to get 20000.

$$\frac{19v^{2}-180v-40}{2000}+20000v=0$$

Divide each term of 19v^{2}-180v-40 by 2000 to get \frac{19}{2000}v^{2}-\frac{9}{100}v-\frac{1}{50}.

$$\frac{19}{2000}v^{2}-\frac{9}{100}v-\frac{1}{50}+20000v=0$$

Combine -\frac{9}{100}v and 20000v to get \frac{1999991}{100}v.

$$\frac{19}{2000}v^{2}+\frac{1999991}{100}v-\frac{1}{50}=0$$

Add \frac{1}{50} to both sides. Anything plus zero gives itself.

$$\frac{19}{2000}v^{2}+\frac{1999991}{100}v=\frac{1}{50}$$

Divide both sides of the equation by \frac{19}{2000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{19}{2000}v^{2}+\frac{1999991}{100}v}{\frac{19}{2000}}=\frac{\frac{1}{50}}{\frac{19}{2000}}$$

Dividing by \frac{19}{2000} undoes the multiplication by \frac{19}{2000}.

$$v^{2}+\frac{\frac{1999991}{100}}{\frac{19}{2000}}v=\frac{\frac{1}{50}}{\frac{19}{2000}}$$

Divide \frac{1999991}{100} by \frac{19}{2000} by multiplying \frac{1999991}{100} by the reciprocal of \frac{19}{2000}.

$$v^{2}+\frac{39999820}{19}v=\frac{\frac{1}{50}}{\frac{19}{2000}}$$

Divide \frac{1}{50} by \frac{19}{2000} by multiplying \frac{1}{50} by the reciprocal of \frac{19}{2000}.

$$v^{2}+\frac{39999820}{19}v=\frac{40}{19}$$

Divide \frac{39999820}{19}, the coefficient of the x term, by 2 to get \frac{19999910}{19}. Then add the square of \frac{19999910}{19} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+\frac{39999820}{19}v+\left(\frac{19999910}{19}\right)^{2}=\frac{40}{19}+\left(\frac{19999910}{19}\right)^{2}$$

Square \frac{19999910}{19} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+\frac{39999820}{19}v+\frac{399996400008100}{361}=\frac{40}{19}+\frac{399996400008100}{361}$$

Add \frac{40}{19} to \frac{399996400008100}{361} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v^{2}+\frac{39999820}{19}v+\frac{399996400008100}{361}=\frac{399996400008860}{361}$$

Factor v^{2}+\frac{39999820}{19}v+\frac{399996400008100}{361}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+\frac{19999910}{19}\right)^{2}=\frac{399996400008860}{361}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{19999910}{19}\right)^{2}}=\sqrt{\frac{399996400008860}{361}}$$

Simplify.

$$v+\frac{19999910}{19}=\frac{2\sqrt{99999100002215}}{19} v+\frac{19999910}{19}=-\frac{2\sqrt{99999100002215}}{19}$$

Subtract \frac{19999910}{19} from both sides of the equation.

$$v=\frac{2\sqrt{99999100002215}-19999910}{19} v=\frac{-2\sqrt{99999100002215}-19999910}{19}$$