$ ext{The value of } \sqrt{20 + \sqrt{20 + \sqrt{20 + \dots \dots \dots \infty}}} ext{ is :}$

Q: $\text{The value of } \sqrt{20 + \sqrt{20 + \sqrt{20 + \dots \dots \dots \infty}}} \text{ is :}$

To solve the problem of finding the value of $\sqrt{20 + \sqrt{20 + \sqrt{20 + \dots \dots \dots \infty}}}$, we need to determine the value of the infinite nested radical expression. Let's denote the entire expression by $x$: \[ x = \sqrt{20 + \sqrt{20 + \sqrt{20 + \dots}}} \] This implies: \[ x = \sqrt{20 + x} \] To eliminate the square root, we square both sides of the equation: \[ x^2 = 20 + x \] Rearranging the equation gives us a quadratic equation: \[ x^2 - x - 20 = 0 \] We solve this quadratic equation using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -1$, and $c = -20$: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} \] \[ x = \frac{1 \pm \sqrt{1 + 80}}{2} \] \[ x = \frac{1 \pm \sqrt{81}}{2} \] \[ x = \frac{1 \pm 9}{2} \] This gives us two potential solutions: \[ x = \frac{1 + 9}{2} = 5 \] \[ x = \frac{1 - 9}{2} = -4 \] Since $x$ represents a length (the value of a square root), it must be non-negative. Therefore, the only valid solution is: \[ x = 5 \] Thus, the value of the infinite nested radical expression is 5, which corresponds to Option 3. **Explanation for Incorrect Options:** 1. **Option 1 (2):** This option is incorrect because solving the quadratic equation derived from the nested radical does not yield 2 as a solution. The correct solution is 5. 2. **Option 2 (4):** This option is incorrect for the same reason as above. The quadratic equation does not provide 4 as a solution. 3. **Option 4 (10):** This option is incorrect because the quadratic equation solution does not include 10. The only valid positive solution is 5. Therefore, the correct answer is Option 3, which is 5.

Question

$	ext{The value of } \sqrt{20 + \sqrt{20 + \sqrt{20 + \dots \dots \dots \infty}}} 	ext{ is :}$

Accepted Answer

To solve the problem of finding the value of $\sqrt{20 + \sqrt{20 + \sqrt{20 + \dots \dots \dots \infty}}}$, we need to determine the value of the infinite nested radical expression.

Let's denote the entire expression by $x$:

$$ x = \sqrt{20 + \sqrt{20 + \sqrt{20 + \dots}}} $$

This implies:

$$ x = \sqrt{20 + x} $$

To eliminate the square root, we square both sides of the equation:

$$ x^2 = 20 + x $$

Rearranging the equation gives us a quadratic equation:

$$ x^2 - x - 20 = 0 $$

We solve this quadratic equation using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -1$, and $c = -20$:

$$ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} $$

$$ x = \frac{1 \pm \sqrt{1 + 80}}{2} $$

$$ x = \frac{1 \pm \sqrt{81}}{2} $$

$$ x = \frac{1 \pm 9}{2} $$

This gives us two potential solutions:

$$ x = \frac{1 + 9}{2} = 5 $$

$$ x = \frac{1 - 9}{2} = -4 $$

Since $x$ represents a length (the value of a square root), it must be non-negative. Therefore, the only valid solution is:

$$ x = 5 $$

Thus, the value of the infinite nested radical expression is 5, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1 (2):** This option is incorrect because solving the quadratic equation derived from the nested radical does not yield 2 as a solution. The correct solution is 5.

2. **Option 2 (4):** This option is incorrect for the same reason as above. The quadratic equation does not provide 4 as a solution.

3. **Option 4 (10):** This option is incorrect because the quadratic equation solution does not include 10. The only valid positive solution is 5.

Therefore, the correct answer is Option 3, which is 5.