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Chapter 3: Problem 191

Solution set of equation \(\mathrm{x}=\sqrt{[12+\sqrt{\\{} 12+\sqrt{(} 12)\\}] \ldots \ldots . . \text { up to }}\) \(\infty\) is (a) 4 (b) \(-4\) (c) 3 (d) \(-3\)

Short Answer

Expert verified
The solution set of the infinite nested radical equation \(x = \sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}\) is (a) 4.

Step by step solution

01

Set up the equation

First, write the given infinite nested equation: \[x = \sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}\] Since this is an infinite nested radical, we can assume that the next term will have the same structure as the whole expression: \[x = \sqrt{12+x}\] #Step 2: Solve for x#
02

Solve for x

To solve for x, square both sides of the equation: \[x^2 = 12+x\] Rearrange the equation to have a quadratic equation: \[x^2 - x - 12 = 0\] Now, we can factor the left side of the equation: \[(x-4)(x+3) = 0\] This gives us two possible solutions for x: \[x = 4, \; -3\] #Step 3: Verify the solutions#
03

Verify the solutions

We must now check if these solutions are valid in the context of our original infinite nested radical equation. If we substitute x = 4 back into the equation, we get: \[4 = \sqrt{12+4}\] \[4 = \sqrt{16}\] \[4 = 4\] This solution is valid. Now, let's substitute x = -3: \[-3 = \sqrt{12-3}\] \[-3 = \sqrt{9}\] However, \(\sqrt{9}\) = 3, not -3, therefore, x = -3 is not a valid solution. After verifying the solutions, we find that the only valid solution from the answer choices is: #Answer# (a) 4

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Most popular questions from this chapter

If the product of roots of equation \(x^{2}-5 k x+4 e^{4 \log k}-3=0\) is 61 then \(\mathrm{k}\) is \(\ldots \ldots \ldots \ldots\) (a) 1 (b) 2 (c) 3 (d) 4

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