Question

Which expression is equivalent to StartFraction 2 a + 1 Over 10 a minus 5 Endfraction divided by StartFraction 10 a Over 4 a squared minus 1 EndFraction?

Answer 1

D

Explanation:

edge 2020

Answer 2

`((2a + 1)^2)/(50a)`

Explanation:

Given

`(2a + 1)/(10a - 5) / (10a)/(4a^2 - 1)`

Required

Find the equivalent

We start by changing the / to *

`(2a + 1)/(10a - 5) / (10a)/(4a^2 - 1)`

`(2a + 1)/(10a - 5) * (4a^2 - 1)/(10a)`

Factorize 10a - 5

`(2a + 1)/(5(2a - 1)) * (4a^2 - 1)/(10a)`

Expand 4a² - 1

`(2a + 1)/(5(2a - 1)) * ((2a)^2 - 1)/(10a)`

`(2a + 1)/(5(2a - 1)) * ((2a)^2 - 1^2)/(10a)`

Express (2a)² - 1² as a difference of two squares

Difference of two squares is such that:
`a^2- b^2= (a+b)(a-b)`

The expression becomes

`(2a + 1)/(5(2a - 1)) * ((2a - 1)(2a + 1))/(10a)`

Combine both fractions to form a single fraction

`((2a + 1)(2a - 1)(2a + 1))/(5(2a - 1)10a)`

Divide the numerator and denominator by 2a - 1

`((2a + 1)((2a + 1))/(5*10a)`

Simplify the numerator

`((2a + 1)^2)/(5*10a)`

`((2a + 1)^2)/(50a)`

Hence,

`(2a + 1)/(10a - 5) / (10a)/(4a^2 - 1)` =
`((2a + 1)^2)/(50a)`