Quadratic Equations
Quadratic Equations
nta_abhyas_2025
Grade 11
Question:
For the equation $|x^2 - 2x - 3| = b$, which of the following statements is true?
For $b = 0$, there are no solutions
For $0 < b < 4$, there are two solutions
For $0 < b < 4$, there are two solutions
For $b = 4$, there are four solutions
Step-by-Step Solution
Key Concept: Use the discriminant of the quadratic equation to determine the number of real solutions based on the parameter $b$.
The intersection points satisfy $x^2 = bx - 4$, or $x^2 - bx + 4 = 0$. The discriminant is $\Delta = b^2 - 16$. When $b < 0$: if $b = 0$, there are no real solutions; if $-4 < b < 0$, then $\Delta < 0$ giving no solutions; if $b = -4$, then $\Delta = 0$ giving one solution (tangent); if $b < -4$, then $\Delta > 0$ giving two distinct solutions.
Correct Answer: 1