Question

$\sqrt{12 + \sqrt{12 + \sqrt{12 ...... = x} } } find the value of x$
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Answer 1

Given:

$\rm \: \sqrt{12 + \sqrt{12 + \sqrt{12 + \sqrt{12 + \sqrt{12 \: ........ \infin} } } } } = x$

To Find :

• The value of x =?

Solution :

As we can see that $\rm{\sqrt{12}}$ is repeating

So, let x be there number repeating in this equation.

$\\ \implies \sf \: \sqrt{12 + \sqrt{12 + \sqrt{12 + \sqrt{12 + \sqrt{12 \: \: .... ....\infin} } } } } = x$

$\\ \implies \sf \: \sqrt{12 + x} = x \\ \\ \implies \sf \: 12 + x = {x}^{2} \\ \\ \implies \sf \: 12 + x - {x}^{2} = 0 \\ \\ \implies \sf \: - {x}^{2} + x + 12 = 0 \\ \\ \implies \sf \: {x}^{2} - x - 12 = 0 \\ \\ \implies \sf \: {x}^{2} - 4x + 3x - 12 = 0 \\ \\ \implies \sf \: x(x - 4) + 3(x - 4) = 0 \\ \\ \implies \sf \: (x + 3)(x - 4) = 0 \\ \\ \implies \sf \: x = 4 \: \: and \: \: x = - 3$

As we know that, negative value is neglected.

So answer will be x=4

$\pink \bigstar\boxed{ \sf{ \sqrt{12 + \sqrt{12 + \sqrt{ 12 + \sqrt{12 \: \: \: \: \: ... .... \infin} } } } = 4}}$