Product against 'Sum-Difference Product'

Algebra Level 3

TRUE or FALSE ?

There exist(s) a pair/pairs of positive real numbers $(a,b)$ with $a > b$ such that their product equals the product of their sum and difference; that is,

$(a+b)(a-b)=ab.$

Inspired from: Agnishom Chattopadhyay .

True False

1 solution

Zain Majumder
Dec 17, 2017

$a^2-b^2=ab$

$a^2-ab+b^2 = 0$

$a = \frac {b \pm \sqrt{b^2+4b^2} } {2} = \frac{b \pm b\sqrt{5}}{2} = \frac{b + b\sqrt{5}}{2}$ since $a$ must be positive.

There are infinite positive solutions for $a$ and $b$ . Therefore, the statement is $true$ .

You could eliminate the minus sign, as both $a$ and $b$ are positive.

Muhammad Rasel Parvej - 3 years, 5 months ago

True! Now fixed.

Zain Majumder - 3 years, 5 months ago

Product against 'Sum-Difference Product'

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