Question

If \[ S=\{m\in \mathbb{R}:x^2-2(1+3m)x+7(3+2m)=0 \text{ has distinct roots}\}, \] then the number of elements in \(S\) is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. Infinite

Hint:

For a quadratic equation to have distinct real roots, always use: \[ D=b^2-4ac\gt 0. \] If the resulting inequality gives intervals, then the set contains infinitely many values.

Answer 1

The Correct Option is D

Step 1: Use the condition for distinct roots.
For a quadratic equation \[ ax^2+bx+c=0 \] to have distinct real roots, its discriminant must be positive.
That is, \[ D\gt 0 \]

Step 2: Identify \(a\), \(b\), and \(c\).
The given equation is \[ x^2-2(1+3m)x+7(3+2m)=0 \] Here, \[ a=1 \] \[ b=-2(1+3m) \] \[ c=7(3+2m) \]

Step 3: Apply the discriminant condition.
\[ D=b^2-4ac \] So, \[ [-2(1+3m)]^2-4(1)\cdot 7(3+2m)\gt 0 \] \[ 4(1+3m)^2-28(3+2m)\gt 0 \]

Step 4: Simplify the inequality.
Dividing by \(4\), we get \[ (1+3m)^2-7(3+2m)\gt 0 \] Now, \[ 1+6m+9m^2-21-14m\gt 0 \] \[ 9m^2-8m-20\gt 0 \]

Step 5: Solve the quadratic inequality.
Factorizing, \[ 9m^2-8m-20\gt 0 \] The roots are \[ m=\frac{8\pm \sqrt{64+720}}{18} \] \[ m=\frac{8\pm 28}{18} \] So, \[ m=2 \] or \[ m=-\frac{10}{9} \] Therefore, \[ 9m^2-8m-20\gt 0 \] when \[ m\lt -\frac{10}{9} \] or \[ m\gt 2 \]

Step 6: Final conclusion.
Thus, \[ S=\left(-\infty,-\frac{10}{9}\right)\cup(2,\infty) \] This set has infinitely many elements.
Therefore, \[ \boxed{\text{Infinite}} \]