The image shows a mathematical problem asking for the value of $x$ given the following infinite nested radical expression:

$x = \sqrt{42 + \sqrt{42 + \sqrt{42 + \sqrt{42 + \dots}}}}.$

The options given are:

• A. 4
• B. 5
• C. 6
• D. 7
• E. 10

Solution:

We can solve this by setting up an equation using the infinite nature of the nested radical:

$x = \sqrt{42 + x}.$

Now, to solve for $x$, square both sides of the equation:

$x^2 = 42 + x.$

This can be rearranged into a quadratic equation:

$x^2 - x - 42 = 0.$

Now, we solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 1$, $b = -1$, and $c = -42$. Plugging in these values:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-42)}}{2(1)} = \frac{1 \pm \sqrt{1 + 168}}{2} = \frac{1 \pm \sqrt{169}}{2}.$

Simplifying further:

$x = \frac{1 \pm 13}{2}.$

This gives us two possible solutions:

$x = \frac{14}{2} = 7, \quad \text{or} \quad x = \frac{-12}{2} = -6.$

Since $x$ represents a length (as it's derived from a square root), we discard the negative solution. Therefore, the correct value of $x$ is:

$\boxed{7}.$

The correct answer is D. 7.

Would you like more details or have any questions?

Related Questions:

1. What happens if the number under the square root changes from 42 to a different value?
2. How would the solution change if the square root structure continued with subtraction instead of addition?
3. Can this method be applied to any other similar infinite nested radical expressions?
4. How do you solve a quadratic equation by factoring instead of using the quadratic formula?
5. What are the implications of negative roots in the context of real-world problems?

Tip:

When dealing with infinite series or nested radicals, try to set up an equation that exploits the repeating structure. This simplifies the problem significantly.