Question

If a, b, c are real and b ≠ 0, can 2b2

– (b+c-a)(a+b-c) be negative, if so give an example.​

Answer 1

Solution.

$\therefore f(a,\:b,\:c)=2b^{2}-(b+c-a)(a+b+c)$

$\quad=2b^{2}-(b+c-a)(b+c+a)$

$\quad=2b^{2}-(b^{2}+c^{2}+2bc-a^{2})$

$\quad=2b^{2}-b^{2}-c^{2}-2bc+a^{2}$

$\quad=a^{2}+b^{2}-c^{2}-2bc$

The coefficient of $c^{2}$ is $(-1)$.

If we put any large $c\in\mathbb{R}$ and smaller $a,\:b\in\mathbb{R}$, the value of the given term can become negative.

An example.

Let, $a=0,\:b=1,\:c=10$

Then, $f(0,\:1,\:10)$

$\quad=0^{2}+1^{2}-10^{2}-2.1.10$

$\quad=0+1-100-20$

$\quad=-119<0$

Note. Many other such $(a,\:b,\:c)$ can be found, just make sure to take large $c\in\mathbb{R}$.

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