Show that, if a, b, c are real numbers and ac = 2(b + d), then at least one of $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots.

Question

Show that, if $a, b, c$ are real numbers and $ac = 2(b + d)$, then, at least one of the equations $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots.

I've have tried many times and used different methods but can't prove it.

Include your attempts into the question – Martund Mar 23 '20 at 10:49 — Martund, Mar 23 '20 at 10:49
$a,b,c$ are real numbers – fine, but what is $d$? – Bernard Mar 23 '20 at 11:06 — Bernard, Mar 23 '20 at 11:06

score 0 · Answer 1 · answered Mar 23 '20 at 10:53

0

Hint:

The sum of discriminants is $$a^2-4b+c^2-4d=a^2+c^2-2ca=(c-a)^2\ge0$$

So, at least one of them must be $\ge0$

answered Mar 23 '20 at 10:53

lab bhattacharjee

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Show that, if a, b, c are real numbers and ac = 2(b + d), then at least one of $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots.

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