What are the real numbers a, b, and c that satisfy certain conditions?

[anemone]

Determine all three real numbers $a, \,b$ and $c$ which satisfies the conditions $a^2+ b^2+ c^2= 26$, $a + b = 5$ and $b + c\ge 7.$

 

[lfdahl]

My attempt:

Spoiler

$a^2+b^2+c^2 = 26\;\;\;\;(1).$

$a+b = 5\;\;\;\;(2).$

$b+c \ge 7\;\;\;\;(3).$

Combining $(1)$ and $(2)$:

\[a^2+b^2+c^2 = (5-b)^2+b^2+c^2 = 26\Rightarrow 2b^2-10b+c^2-1 = 0, \;\;\;\;(4).\]

From $(3)$ one gets: \[c^2 \geq (7-b)^2, \;\;\;\; (5).\]

Combining $(4)$ and $(5)$:

\[2b^2-10b+c^2-1 = 0 \geq 2b^2-10b+(7-b)^2-1 = 3b^2-24b+48\]

\[\Rightarrow b^2-8b+16 \leq 0\]

Or \[(b-4)^2 \leq 0\]

So the only possible $b$-value is $4$. Thus $a = 1$ (from $(2)$) and $c = 3$ (from $(1)$ and $(3)$).