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If a = √17 + 4, find the value of (3a2 – 5a – 3)/(4a2 – 3a – 4)
1. 22/17
2. 25/16
3. 15/16
4. 19/29

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Correct Answer - Option 4 : 19/29

Given:

a = √17 + 4

Concept Used:

(a + b) × (a - b) = a2 - b2

Concept of rationalisation: When an irrational number came in denominator in a fraction then it become difficult in calculation. To avoid this problem we multiply something with both numerator and denominator so that the denominator part become rational. This process is known as rationalisation.

In general, the process to make an irrational number to a rational number is called rationalisation.

Calculation:

a = √17 + 4

⇒ 1/a = 1/(√17 + 4)

By rationalisation 

⇒ \(\frac{1}{a} = \frac{1}{{√ {17} + 4}} × \frac{{√ {17} - 4}}{{√ {17} - 4}}\)

⇒ 1/a = (√17 – 4)/(17 – 16)

⇒ 1/a = √17 – 4      ----1

⇒ a – 1/a = √17 + 4 – √17 + 4

⇒ a – 1/a = 8      ----2

To find (3a2 – 5a – 3)/(4a2 – 3a – 4)

⇒ (3a2 – 5a – 3)/(4a2 – 3a – 4)

Divide numerator and denominator by a

⇒ (3a – 5 – 3/a)/(4a – 3 – 4/a)

⇒ [3(a – 1/a)  5]/[4(a  1/a)  3]

⇒ (3 × 8  5)/(4 × 8  3)    (from equation 2)

⇒ 19/29

∴ The value of (3a2 – 5a – 3)/(4a2 – 3a – 4) is 19/29

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