The value of $\sqrt{72 + \sqrt{72 + \sqrt{72 + \dots \infty}}}$ is :

Q: The value of $\sqrt{72 + \sqrt{72 + \sqrt{72 + \dots \infty}}}$ is :

To solve the problem of finding the value of $\sqrt{72 + \sqrt{72 + \sqrt{72 + \dots \infty}}}$, we need to understand that this expression represents an infinite nested radical. Let's denote the entire expression by $x$: \[ x = \sqrt{72 + \sqrt{72 + \sqrt{72 + \dots}}} \] This implies: \[ x = \sqrt{72 + x} \] To solve for $x$, we square both sides of the equation: \[ x^2 = 72 + x \] Rearranging the equation gives us: \[ x^2 - x - 72 = 0 \] This is a quadratic equation, and we can solve it using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = 1$, $b = -1$, and $c = -72$. Plugging these values into the formula gives: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \] \[ x = \frac{1 \pm \sqrt{1 + 288}}{2} \] \[ x = \frac{1 \pm \sqrt{289}}{2} \] \[ x = \frac{1 \pm 17}{2} \] This results in two potential solutions: \[ x = \frac{18}{2} = 9 \] \[ x = \frac{-16}{2} = -8 \] Since $x$ represents a length (as it is derived from a square root), it must be non-negative. Therefore, the only valid solution is: \[ x = 9 \] Thus, the value of $\sqrt{72 + \sqrt{72 + \sqrt{72 + \dots \infty}}}$ is 9, which corresponds to Option 3. **Explanation of why other options are incorrect:** 1. **Option 1 (6):** If $x = 6$, substituting into the equation $x = \sqrt{72 + x}$ gives $6 = \sqrt{72 + 6}$, which simplifies to $6 = \sqrt{78}$. This is incorrect because $\sqrt{78}$ is not equal to 6. 2. **Option 2 (8):** If $x = 8$, substituting into the equation gives $8 = \sqrt{72 + 8}$, which simplifies to $8 = \sqrt{80}$. This is incorrect because $\sqrt{80}$ is not equal to 8. 3. **Option 4 (12):** If $x = 12$, substituting into the equation gives $12 = \sqrt{72 + 12}$, which simplifies to $12 = \sqrt{84}$. This is incorrect because $\sqrt{84}$ is not equal to 12. Therefore, the only correct answer is Option 3, which is 9.

Question

Accepted Answer

To solve the problem of finding the value of $\sqrt{72 + \sqrt{72 + \sqrt{72 + \dots \infty}}}$, we need to understand that this expression represents an infinite nested radical. Let's denote the entire expression by $x$:

$$ x = \sqrt{72 + \sqrt{72 + \sqrt{72 + \dots}}} $$

This implies:

$$ x = \sqrt{72 + x} $$

To solve for $x$, we square both sides of the equation:

$$ x^2 = 72 + x $$

Rearranging the equation gives us:

$$ x^2 - x - 72 = 0 $$

This is a quadratic equation, and we can solve it using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

where $a = 1$, $b = -1$, and $c = -72$. Plugging these values into the formula gives:

$$ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} $$

$$ x = \frac{1 \pm \sqrt{1 + 288}}{2} $$

$$ x = \frac{1 \pm \sqrt{289}}{2} $$

$$ x = \frac{1 \pm 17}{2} $$

This results in two potential solutions:

$$ x = \frac{18}{2} = 9 $$

$$ x = \frac{-16}{2} = -8 $$

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

$$ x = 9 $$

Thus, the value of $\sqrt{72 + \sqrt{72 + \sqrt{72 + \dots \infty}}}$ is 9, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

1. **Option 1 (6):** If $x = 6$, substituting into the equation $x = \sqrt{72 + x}$ gives $6 = \sqrt{72 + 6}$, which simplifies to $6 = \sqrt{78}$. This is incorrect because $\sqrt{78}$ is not equal to 6.

2. **Option 2 (8):** If $x = 8$, substituting into the equation gives $8 = \sqrt{72 + 8}$, which simplifies to $8 = \sqrt{80}$. This is incorrect because $\sqrt{80}$ is not equal to 8.

3. **Option 4 (12):** If $x = 12$, substituting into the equation gives $12 = \sqrt{72 + 12}$, which simplifies to $12 = \sqrt{84}$. This is incorrect because $\sqrt{84}$ is not equal to 12.

Therefore, the only correct answer is Option 3, which is 9.