Question

If a= 3_2✓2 then find a^2-1/a^2

Answer 1

Question:

If a = 3 - 2√2 then find a² - 1/a².

Solution:

First we have a look at what  1/a  is!

1/a = 1/(3 - 2√2)

1/a = (3 + 2√2)/[(3 - 2√2)(3 + 2√2)]

1/a = (3 + 2√2)/(3² - (2√2)²)

1/a = (3 + 2√2)/(9 - 8)

1/a = 3 + 2√2

So,

a - 1/a = (3 - 2√2) - (3 + 2√2)

a - 1/a = 3 - 2√2 - 3 - 2√2

a - 1/a = - 4√2

a + 1/a = (3 - 2√2) + (3 + 2√2)

a + 1/a = 3 - 2√2 + 3 + 2√2

a + 1/a = 6

Now, we find  a² - 1/a².

a² - 1/a² = (a - 1/a)(a + 1/a)

a² - 1/a² = (- 4√2) · 6

a² - 1/a² = - 24√2

Hence,  -24√2  is the answer.