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

• A. 7
• B. 8
• C. 6
• D. 9

Hint:

Rewrite the quadratic equation in a form that allows you to find the integer roots easily.

Answer 1

The Correct Option is C

1. Rewrite the equation: \[ x^2 + 4x + 4 = n + 4 \] \[ (x + 2)^2 = n + 4 \]
2. Solve for $x$: \[ x = -2 \pm \sqrt{n + 4} \]
3. Determine the range of $n$: \[ 20 \leq n \leq 100 \] \[ \sqrt{24} \leq \sqrt{n + 4} \leq \sqrt{104} \] \[ 4.9 \leq \sqrt{n + 4} \leq 10.2 \]
4. Find the integer values of $\sqrt{n + 4}$: \[ \sqrt{n + 4} \in \{5, 6, 7, 8, 9, 10\} \] 5. Calculate the number of distinct values of $n$: \[ \text{Number of distinct values} = 6 \] Therefore, the correct answer is (3) 6.