Question

A=2+root 3/2-root3 b=2-root3/2+root3 find a^2+b^2+ab

Answer 1

Answer:

Given

A = 2 + √3/2 - √3

B = 2 - √3/2 + √3

To find

A² + B² + AB

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

Using

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

= 2² + 4√3 + (√3)²/2² - 4√3 + (√3)² + 2² - 4√3 + (√3)²/2² + 4√3 + (√3)² + 1

= 7 + 4√3/7 - 4√3 + 7 - 4√3/7 + 4√3 + 1

= (7 + 4√3)² + (7 - 4√3)²/(7 + 4√3)(7 - 4√3) + 1

Using

(a + b)(a - b) = a² - b²

= 7² + 8√3 + (4√3)² + 7² - 8√3 + (4√3)²/7² - (4√3)² + 1

= 49 + 48 + 49 + 48/49 - 48 + 1

= 98 + 96 /1 + 1

= 194 + 1

= 195

Concepts used:

→ Surds

More Information:

→ (a + b)² - (a - b)² = 4ab

→ (a + b)² + (a - b)² = 2(a² + b²)