Answer:
[tex]\frac{(2a + 1)^2}{50a}[/tex]
Step-by-step explanation:
Given
[tex]\frac{2a + 1}{10a - 5} / \frac{10a}{4a^2 - 1}[/tex]
Required
Find the equivalent
We start by changing the / to *
[tex]\frac{2a + 1}{10a - 5} / \frac{10a}{4a^2 - 1}[/tex]
[tex]\frac{2a + 1}{10a - 5} * \frac{4a^2 - 1}{10a}[/tex]
Factorize 10a - 5
[tex]\frac{2a + 1}{5(2a - 1)} * \frac{4a^2 - 1}{10a}[/tex]
Expand 4a² - 1
[tex]\frac{2a + 1}{5(2a - 1)} * \frac{(2a)^2 - 1}{10a}[/tex]
[tex]\frac{2a + 1}{5(2a - 1)} * \frac{(2a)^2 - 1^2}{10a}[/tex]
Express (2a)² - 1² as a difference of two squares
Difference of two squares is such that: [tex]a^2- b^2= (a+b)(a-b)[/tex]
The expression becomes
[tex]\frac{2a + 1}{5(2a - 1)} * \frac{(2a - 1)(2a + 1)}{10a}[/tex]
Combine both fractions to form a single fraction
[tex]\frac{(2a + 1)(2a - 1)(2a + 1)}{5(2a - 1)10a}[/tex]
Divide the numerator and denominator by 2a - 1
[tex]\frac{(2a + 1)((2a + 1)}{5*10a}[/tex]
Simplify the numerator
[tex]\frac{(2a + 1)^2}{5*10a}[/tex]
[tex]\frac{(2a + 1)^2}{50a}[/tex]
Hence,
[tex]\frac{2a + 1}{10a - 5} / \frac{10a}{4a^2 - 1}[/tex] = [tex]\frac{(2a + 1)^2}{50a}[/tex]
