Consider the equation x² + 4x - n = 0, where n ∈ [20, 100] is a natural number. Then the number of a

Q: Why can't we just use the discriminant $D > 0$?

The condition $D > 0$ only guarantees that the roots are real and distinct. It does not guarantee that they are integers. For the roots to be integers, the discriminant must be a perfect square, which is a much stronger condition.

Q: Wait, why do we reject the negative root here?

We don't actually reject the negative root! The quadratic formula gives us two roots: $x = -2 + \sqrt{4+n}$ and $x = -2 - \sqrt{4+n}$. Both are valid roots. The question asks for the number of values of $n$ for which the equation has integral roots. As long as $\sqrt{4+n}$ is an integer, both roots will be integers.

Q: Is it possible for $n$ to be a value that makes $4+n$ a perfect square but not a natural number?

The problem states $n \in [20, 100]$ and $n \in \mathbb{N}$. Since our perfect squares are all greater than $24$, $n = k^2 - 4$ will always be greater than $20$, which is a natural number. So, all our found values of $n$ are valid.

Question

Consider the equation $x^2 + 4x - n = 0$, where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

Options:

(A) 7

(B) 8

(C) 6

(D) 5

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the number of natural values of $n$ in the range $20 \le n \le 100$ such that the quadratic equation $x^2 + 4x - n = 0$ possesses integral roots.

To solve this, we employ the quadratic formula:

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

Given our coefficients $a = 1$, $b = 4$, and $c = -n$, we substitute these values into the formula to obtain:

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

### The Master Equation

Simplifying the expression under the radical, we get:

$$x = \frac{-4 \pm \sqrt{16 + 4n}}{2}$$

By factoring out a $4$ from the discriminant, the expression becomes:

$$x = \frac{-4 \pm 2\sqrt{4 + n}}{2}$$

Dividing by $2$, we arrive at the simplified form:

$$x = -2 \pm \sqrt{4 + n}$$

### Establishing the Condition

For $x$ to be an integer, the term $\sqrt{4 + n}$ must necessarily be an integer. This implies that $4 + n$ must be a **perfect square**.

Let $4 + n = k^2$ for some integer $k$. Given the constraint $20 \le n \le 100$, we add $4$ to all parts of the inequality to bound $k^2$:

$$24 \le 4 + n \le 104$$

$$24 \le k^2 \le 104$$

### Final Calculation

We identify all perfect squares $k^2$ that fall within the interval $[24, 104]$:

*   $5^2 = 25$
*   $6^2 = 36$
*   $7^2 = 49$
*   $8^2 = 64$
*   $9^2 = 81$
*   $10^2 = 100$

For each of these $6$ values of $k^2$, we calculate the corresponding $n$ using $n = k^2 - 4$:

*   $n = 25 - 4 = 21$
*   $n = 36 - 4 = 32$
*   $n = 49 - 4 = 45$
*   $n = 64 - 4 = 60$
*   $n = 81 - 4 = 77$
*   $n = 100 - 4 = 96$

Since all these values of $n$ are natural numbers within the specified range, there are exactly **$6$** such values.

Animated Solution for Mathematics - Quadratic Equations: Consider the equation $x^{2} + 4 x - n = 0$ , where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$ , for which the given equation has integral roots, is equal to

Select Answer:

Visualized Solution

Analyze the Equation

Apply the Quadratic Formula

Substitute and Simplify

Final Root Expression

Condition for Integral Roots

Determine the Range of $4 + n$

Identify Perfect Squares in Range

Final Conclusion

The Sigma Insight: Nature of Roots

Analyzing the Setup

The Master Equation

Establishing the Condition

Final Calculation

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