Question

If a= 1/1+√2 then find the value of a³+2a²-3a+5​

Answer 1

Answer :

a³ + 2a² - 3a + 5 = 7 - 2√5

Solution :

• Given : a = 1/(1 + √2)
• To find : a³ + 2a² - 3a + 5 = ?

We have ;

=> a = 1/(1 + √2)

=> a = 1/(√2 + 1)

Now ,

Rationalising the denominator of the term in RHS , we have ;

=> a = (√2 - 1)/(√2 + 1)(√2 - 1)

=> a = (√2 - 1)/[ (√2)² - 1² ]

=> a = (√2 - 1)/(2 - 1)

=> a = (√2 - 1)/1

=> a = √2 - 1

=> a + 1 = √2

Now ,

Squaring both the sides , we have ;

=> (a + 1)² = (√2)²

=> a² + 2a + 1 = 2

=> a² + 2a + 1 - 2 = 0

=> a² + 2a - 1 = 0

Now ,

=> a³ + 2a² - 3a + 5 = a³ + 2a² - a - 2a + 5

=> a³ + 2a² - 3a + 5 = a(a² + 2a - 1) - 2a + 5

=> a³ + 2a² - 3a + 5 = a•0 - 2a + 5

=> a³ + 2a² - 3a + 5 = 0 - 2(√2 - 1) + 5

=> a³ + 2a² - 3a + 5 = -2√2 + 2 + 5

=> a³ + 2a² - 3a + 5 = 7 - 2√5

Hence ,

a³ + 2a² - 3a + 5 = 7 - 2√5