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Simplify this equation: 25-a^2 / 3a^2-13a-10

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rosey-girl | Student, Grade 12 | Valedictorian

Posted May 15, 2013 at 9:36 PM via web

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Simplify this equation:

25-a^2 / 3a^2-13a-10

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violy | High School Teacher | (Level 3) Assistant Educator

Posted May 15, 2013 at 10:31 PM (Answer #1)

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We can factor the top using Difference of Two Squares. 

`x^2 - y^2 = (x - y)(x + y)`

`25 - a^2 = (5 - a)(5 + a)`

For the bottom, we can use ac method. 

First, identify a, b, and c. 

a = 3, b = -13 and c = -10. 

Multiply a and c. 3 * -10 = -30. 

We will find two numbers which when we multiply we get -30 (value of ac), and when we add we get -13 (value of b).

We can first lists the pair factors of -30 and their sum.

-30 and 1 :   -30 + 1 = -29

-10 and 3:  -10 + 3 = -7

-6 and 5:  -6 + 5 = -1.

-15 and 2:  -15 + 2 = -13.

Hence, the numbers we are looking for are -15 and 2. We will use those numbers to rewrite the middle term.

`3a^2 - 15a + 2a - 10`

We can split the terms into two groups.

`(3a^2 -15a) + (2a - 10)`

Factor out the gcf from each group.

` ` `3a(a - 5) + 2(a - 5)`

Factor out the common factor.

`(a - 5)(3a + 2)`

We can factor out a -1 from a - 5. 

`-(-a + 5)(3a + 2) = -(5 - a)(3a + 2)`

So, our rational expression will be: 

`((5 - a)(5 + a))/(-(5 - a)(3a + 2))`

Cancel common factor on top and bottom.

So, the answer will be: -(a + 5)/(3a + 2) .

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