Solution 2.1:5a
From Förberedande kurs i matematik 1
(Difference between revisions)
m (Lösning 2.1:5a moved to Solution 2.1:5a: Robot: moved page) |
m |
||
| (One intermediate revision not shown.) | |||
| Line 1: | Line 1: | ||
| - | { | + | In the same way that we calculated fractions, we can subtract the terms' numerators if we first expand the fractions so that they have the same denominator. Because the denominators are <math>x-x^{2}=x(1-x)</math> and <math>x</math>, the lowest common denominator is <math>x(1-x)</math>, |
| - | < | + | |
| - | {{ | + | {{Displayed math||<math>\begin{align} |
| + | \frac{1}{x-x^{2}}-\frac{1}{x\vphantom{x^2}} | ||
| + | &= \frac{1}{x-x^{2}}-\frac{1}{x\vphantom{x^2}}\cdot \frac{1-x}{1-x\vphantom{x^2}}\\[5pt] | ||
| + | &= \frac{1}{x-x^{2}}-\frac{1-x}{x-x^{2}}\\[5pt] | ||
| + | &= \frac{1-(1-x)}{x-x^{2}}\\[5pt] | ||
| + | &= \frac{1-1+x}{x-x^{2}}\\[5pt] | ||
| + | &= \frac{x}{x-x^{2}}\,\textrm{.} | ||
| + | \end{align}</math>}} | ||
| + | |||
| + | This fraction can be simplified by eliminating the factor ''x'' from the numerator and denominator | ||
| + | |||
| + | {{Displayed math||<math>\frac{x}{x-x^{2}} = \frac{x}{x(1-x)} = \frac{1}{1-x}\,\textrm{.}</math>}} | ||
Current revision
In the same way that we calculated fractions, we can subtract the terms' numerators if we first expand the fractions so that they have the same denominator. Because the denominators are \displaystyle x-x^{2}=x(1-x) and \displaystyle x, the lowest common denominator is \displaystyle x(1-x),
| \displaystyle \begin{align}
\frac{1}{x-x^{2}}-\frac{1}{x\vphantom{x^2}} &= \frac{1}{x-x^{2}}-\frac{1}{x\vphantom{x^2}}\cdot \frac{1-x}{1-x\vphantom{x^2}}\\[5pt] &= \frac{1}{x-x^{2}}-\frac{1-x}{x-x^{2}}\\[5pt] &= \frac{1-(1-x)}{x-x^{2}}\\[5pt] &= \frac{1-1+x}{x-x^{2}}\\[5pt] &= \frac{x}{x-x^{2}}\,\textrm{.} \end{align} |
This fraction can be simplified by eliminating the factor x from the numerator and denominator
| \displaystyle \frac{x}{x-x^{2}} = \frac{x}{x(1-x)} = \frac{1}{1-x}\,\textrm{.} |
