Find the value of a in (3 - √5) / (3 + 2√5) = a√5 - 19/11

Solution:

Given, the expression is (3 - √5) / (3 + 2√5) = a√5 - 19/11

We have to find the value of a.

Considering LHS,

LHS: (3 - √5) / (3 + 2√5)

By taking conjugate,

(3 - √5) / (3 + 2√5) = (3 - √5) / (3 + 2√5) × (3 - 2√5) / (3 - 2√5)

= (3 - √5)(3 - 2√5) / (3 + 2√5)(3 - 2√5)

(a² - b²) = (a - b)(a + b)

(3 + 2√5)(3 - 2√5) = (3)² - (2√5)²

= 9 - 4(5)

= 9 - 20

= -11

So, (3 - √5)(3 - 2√5) / (3 + 2√5)(3 - 2√5) = (3 - √5)(3 - 2√5) / (-11)

By multiplicative and distributive property,

(3 - √5)(3 - 2√5) = 3(3) - 3(2√5) - 3(√5) + 2√5(√5)

= 9 - 6√5 - 3√5 + 2(5)

= 9 - 9√5 + 10

= 19 - 9√5

Now, (3 - √5)(3 - 2√5) / (-11) = (19 - 9√5) / (-11)

= 19/(-11) - 9√5/(-11)

= 9√5/11 - 19/11

So, a√5 - 19/11 = 9√5/11 - 19/11

a√5 - 19/11 + 19/11 = 9√5/11

a√5 = 9√5/11

a = 9/11

Therefore, the value of a is 9/11

✦ Try This: Find the values of a³ + b³ if a = 1/(7 - 4√3) and b = 1/(7 + 4√3)

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1

NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 11(ii)

Summary:

The surd that is used to multiply is called the rationalizing factor (RF). The value of a in (3 - √5) / (3 + 2√5) = a√5 - 19/11 is 9/11

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