Key Concepts and Formulas

• Quadratic Formula: For a quadratic equation $ax^2 + bx + c = 0$, the roots are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Discriminant: The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $\Delta = b^2 - 4ac$. The quadratic equation has integer roots if and only if the discriminant is a perfect square, and the roots obtained using the quadratic formula are integers.
• Perfect Square: A perfect square is an integer that can be expressed as the square of another integer.

Step-by-Step Solution

1. Applying the Quadratic Formula

We are given the equation $x^2 + 4x - n = 0$. We want to find the roots of this equation. We apply the quadratic formula with $a=1$, $b=4$, and $c=-n$: $x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-n)}}{2(1)}$ $x = \frac{-4 \pm \sqrt{16 + 4n}}{2}$

• Explanation: We use the quadratic formula to express the roots of the given quadratic equation in terms of $n$. This is a direct application of a fundamental formula.

2. Simplifying the Expression for Roots

To make it easier to determine when the roots are integers, we simplify the expression: $x = \frac{-4 \pm \sqrt{4(4 + n)}}{2}$ We can take $\sqrt{4}$ out of the square root: $x = \frac{-4 \pm 2\sqrt{4 + n}}{2}$ Now, we can divide both terms in the numerator by 2: $x = -2 \pm \sqrt{4 + n}$

• Explanation: This simplification isolates the term that needs to be an integer for the roots to be integers.

3. Determining the Condition for Integral Roots

For the roots $x$ to be integers, the term $\sqrt{4+n}$ must be an integer. Since $-2$ is an integer, adding or subtracting an integer from it will always result in an integer. Therefore, we need $4+n$ to be a perfect square. We can write this as: $4+n = k^2$ where $k$ is a non-negative integer.

• Explanation: This step converts the problem into finding values of $n$ such that $4+n$ is a perfect square.

4. Analyzing the Range for 'n' and '4+n'

We are given that $20 \le n \le 100$. We need to find the range for $4+n$: $20 + 4 \le n + 4 \le 100 + 4$ $24 \le 4+n \le 104$ So, we are looking for perfect squares ($k^2$) in the range $[24, 104]$.

• Explanation: This is an important step as it defines the interval where we need to find perfect squares.

5. Identifying Perfect Squares within the Range

We list the perfect squares ($k^2$) that satisfy $24 \le k^2 \le 104$:

• The smallest integer whose square is at least 24 is 5 ($5^2 = 25$).
• The perfect squares are:
  • $5^2 = 25$
  • $6^2 = 36$
  • $7^2 = 49$
  • $8^2 = 64$
  • $9^2 = 81$
  • $10^2 = 100$
• The next perfect square, $11^2 = 121$, is greater than 104, so it is outside our range.

• Explanation: We are finding values of k such that $k^2$ is in our range.

6. Finding the Corresponding Values of 'n'

For each perfect square $k^2$ found, we can determine the value of $n$ using the relationship $n = k^2 - 4$. We also verify that $20 \le n \le 100$.

• If $4+n = 25$, then $n = 25 - 4 = 21$. ($20 \le 21 \le 100$, valid)
• If $4+n = 36$, then $n = 36 - 4 = 32$. ($20 \le 32 \le 100$, valid)
• If $4+n = 49$, then $n = 49 - 4 = 45$. ($20 \le 45 \le 100$, valid)
• If $4+n = 64$, then $n = 64 - 4 = 60$. ($20 \le 60 \le 100$, valid)
• If $4+n = 81$, then $n = 81 - 4 = 77$. ($20 \le 77 \le 100$, valid)
• If $4+n = 100$, then $n = 100 - 4 = 96$. ($20 \le 96 \le 100$, valid)

• Explanation: This step calculates the values of $n$ corresponding to the perfect squares found in the previous step and verifies that these values fall within the given range.

7. Counting the Distinct Values of 'n'

We have identified the following distinct values of $n$ that satisfy all conditions: $\{21, 32, 45, 60, 77, 96\}$. There are a total of 6 such distinct values.

• Explanation: We count the number of values of $n$ that satisfy all the conditions to arrive at the final answer.

Common Mistakes & Tips

• Tip: Always check that the calculated values of $n$ lie within the specified interval $[20, 100]$.
• Common Mistake: Incorrectly defining the range for $4+n$.
• Tip: Remember to check that $\sqrt{4+n}$ yields an integer value.

Summary

We used the quadratic formula to find the roots of the equation $x^2 + 4x - n = 0$. We then simplified the expression for the roots and determined that $4+n$ must be a perfect square for the roots to be integers. We found the range for $4+n$ to be $[24, 104]$ and then found all perfect squares in this range. These perfect squares corresponded to 6 distinct values of $n$ within the given range.

Final Answer

The final answer is $\boxed{6}$, which corresponds to option (A).