If (a + b) = 2 and (a - b) = 10, find the values of:
(i) (a² + b²)
(ii) ab
Use the formulas (a + b)² + (a - b)² = 2(a² + b²) and (a + b)² - (a - b)² = 4ab
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Answer:
Given,Simplify (a + b) ² - (a - b) ²?
So first,
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a + b)² – (a – b)² = (a² + 2ab + b²) – (a² – 2ab + b²)
While subtracting polynomials, the signs in the first polynomial are unchanged when the parentheses are dropped, whereas the middle – sign distributes to all of the terms in the second polynomial, thus changing all the signs in the second polynomial when the parentheses are dropped
(a² + 2ab + b²) – (a² – 2ab + b²) = a² + 2ab + b² – a² + 2ab – b²
Rearrange the terms and you get:
a² – a² + 2ab + 2ab + b² – b²
Cancel out same terms with opposite signs, and you get:
2ab + 2ab
which simplifies to our final answer:
4ab
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