Question : Find the value of $\sqrt{30+\sqrt{30+\sqrt{30+...}}}$ :
Option 1: 5
Option 2: $3\sqrt{10}$
Option 3: 6
Option 4: 7
Correct Answer: 6
Solution :
Let $x = \sqrt{30+\sqrt{30+\sqrt{30+...}}}$
So, ${x = }\sqrt{30+x}$
By squaring both sides, we get,
$⇒x^{2}=(\sqrt{30+x})^{2}$
$⇒x^{2}=30+x$
$⇒x^{2}–x–30=0$
$⇒x^{2}– 6x+5x–30=0$
$⇒x(x–6)+5(x–6) = 0$
$⇒(x–6)(x+5)=0$
$⇒x=6,–5$
A square root cannot be negative. Thus, $x$ cannot be –5.
So, $x = 6$
Hence, the correct answer is 6.
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