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If $$a = \sqrt{23a + b}$$, $$b = \sqrt{23b + a}$$, $$a \neq b$$, then the value of $$\sqrt{a^2 + b^2 + 48}$$ is
Squaring the equations we get
$$a^2 = 23a + b$$ and $$b^2 = 23b + a$$.
Subtracting the two gives us
$$a^2 - b^2 = 22(a-b) \implies (a-b)(a+b) = 22(a-b) $$
As $$a \neq b$$ this gives
$$a + b = 22$$.
Adding the two equations we get,
$$a^2 + b^2 = 24(a + b) = 528$$
$$\sqrt{a^2 + b^2 + 48} = \sqrt{576} = 24$$.
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