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Solving a Quadratic Equation by Completion of Squares Method

Prove that: ...

Question

Prove that:
$(a^{- 1} + b^{- 1})^{- 1} = \frac{a b}{a + b}$

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Solution

$a^{- 1} + b^{- 1} = \frac{1}{a} + \frac{1}{b} = \frac{a + b}{a b}$

So, $(a^{- 1} + b^{- 1})^{- 1} = \frac{a b}{a + b}$

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Prove that

(a^-1 + b^-1)^-1 = ab/a+b

Q. Let a and b be positive real numbers such that

a + b = 1

a^{a} b^{b} + a^{a} b^{b} \leq 1

Q. If A, B are two matrix then prove that

(A B)^{- 1} = B^{- 1} A^{- 1}

a > b > c > 0

then prove that

{cot}^{- 1} (\frac{a b + 1}{a - b}) + {cot}^{- 1} (\frac{b c + 1}{b - c}) + {cot}^{- 1} (\frac{c a + 1}{c - a}) = π

Q. If 1/a, 1/b, 1/c, are in a.p prove that BC, ca, an are in a.p
hints: prove that ab- ca =ca- BC
Using 1/c - 1/b = 1/b - 1/a.

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