Answer:
(7 + 4√3)² + (7 - 4√3)² = 194
Step-by-step explanation:
* Lets explain how to solve the problem
∵ a² + b² = (7 + 4√3)² + (7 - 4√3)²
- Lets solve each bracket alone then add the two answers
# Remember (a + b)² = (a)² + 2(a)(b) + (b)²
∵ (7 + 4√3)² = (7)² + 2(7)(4√3) + (4√3)²
∵ 7² = 49
∵ 2(7)(4√3) = 2 × 7 × 4 × √3 = 56√3
∵ (4√3)² = (4)² × (√3)² = 16 × 3 = 48
∴ (7 + 4√3)² = 49 + 56√3 + 48
- 49 and 48 are like terms
∴ (7 + 4√3)² = 97 + 56√3 ⇒ (1)
# Remember (a - b)² = (a)² - 2(a)(b) + (b)²
∵ (7 - 4√3)² = (7)² - 2(7)(4√3) + (4√3)²
- All the numbers like the first bracket except the middle sign, then
all the answer will be the same except the middle sign will be (-)
∴ (7 - 4√3)² = 49 - 56√3 + 48
∴ (7 - 4√3)² = 97 - 56√3 ⇒ (2)
- lets add answers (1) and (2)
∴ (7 + 4√3)² + (7 - 4√3)² = 97 + 56√3 + 97 - 56√3
∴ (7 + 4√3)² + (7 - 4√3)² = (97 + 97) + (56√3 - 56√3)
∵ 97 + 97 = 194
∵ 56√3 - 56√3 = 0
∴ (7 + 4√3)² + (7 - 4√3)² = 194
⇒ expanding the bracket gives,
⇒ rearranging to equate one side to zero
⇒ Dividing each term by 7 to simplify
⇒ Factorising gives
and 