What is the simplified form of 3/(2x + 5) + 21/(8x² + 26x + 15)

Solution:

Step 1:

First we need to factorise 8x² + 26x + 15 so that LCM is possible to take for both the terms.

On factorising the term 8x² + 26x + 15 by factorization method

The factors of product 120(8 × 15) are 20 × 6

So that 20 + 6 gives a middle term that is 26 which is the rule of factorization of quadratic equation

8x² + 26x + 15 = 8x² + 20x + 6x +15

= 4x(2x + 5) + 3(2x + 5)

= (2x + 5)(4x + 3)

Step2:

 ⇒3/(2x + 5) + 21/(8x² + 26x + 15) = 3/(2x + 5) + 21/(2x + 5)(4x + 3)

In the first term only 2x + 5 is the term in the denominator which is need to multiplied by (4x + 3) with both numerator and denominator

⇒3/(2x + 5) + 21/(2x + 5)(4x + 3) = [3(4x + 3)/(4x + 3)(2x+5)] + [21/(2x + 5)(4x + 3)]

Step3:

After taking LCM for both the terms

⇒ 3/(2x + 5) + 21/(2x + 5)(4x + 3) = [3(4x + 3) + 21 ]/(4x + 3)(2x + 5)

⇒ 3/(2x + 5) + 21/(2x + 5)(4x + 3) =[12x + 9 + 21]/(4x + 3)(2x + 5)

⇒ 3/(2x + 5) + 21/(2x + 5)(4x + 3) =[6x + 30]/(4x + 3)(2x + 5)

⇒ 3/(2x + 5) + 21/(2x + 5)(4x + 3) = 6(x + 5)/(4x + 3)(2x + 5)

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What is the simplified form of 3/(2x + 5) + 21/(8x² + 26x + 15)

Summary :

The simplified form of 3/(4x + 3) + 21/(8x² + 26x + 15) is 6(x + 5)/(4x + 3)(2x + 5) .

Since it is not possible to do further simplification 6(x + 5)/(4x + 3)(2x + 5) is the simplified form of the given expression.