Question

a+b+c=14
a^2+b^2+c^2=84
b^2=ac
solve for integer values , tell the highest integer value among them

Answer 1

\[a+b+c=14\]
\[a^2+b^2+c^2=84\]
\[b^2=ac\]

Answer 2

You should know an identity here:\[(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac)\]Use that to get:\[(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + b^2)\]\[(14)^2 = 84 + 2b(a + b + c) = 84 + 28b\]\[b = \frac{196 - 84}{28} = 4\]

Can you get the rest? :)

Answer 3

@AkashdeepDeb Nice Approach..

Answer 4

have u used\[(a+b+c)^2=((a+b)+c))^2\]
ie
\[(a+b)^2=a^2+2ab+b^2\]

Answer 5

substititing\[a+b=variable\]

Answer 6

Yes, you get the formula I posted there, just like that.

Answer 7

Group a and b together and then use that formula, expand it you will get the same..

Answer 8

nope its a property it comes for three or more variables as ...
(a+b+c)^2 = ((a+b) + c)^2
=> (a+b)^2 + (c^2) + 2(a+b)*c
=> a^2 + b^2 + c^2 + 2ab +2ac +2bc @mathmath333

Answer 9

\[((a+b) +c )^2 = (a+b)^2 + c^2 + 2(a+b) \cdot c \\ \implies a^2 + b^2 + 2ab + c^2 + 2(a+b) c \\ \implies a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\\ \implies \color{green}{a^2 + b^2 + c^2 + 2(ab + bc + ac)}\]

Answer 10

thanku all