10.
Which of the following statements about the function f(x) = x2 − 3x + 4 is true?
a. The function does not intersect the x-axis
b. The function does not intersect the y-axis
c. The function has two real roots
d. The function has one real and one imaginary root
f(x) = x² – 3x + 4 Does Not Intersect X-Axis | Discriminant Analysis
Correct Answer
f(x) = x² – 3x + 4 does not intersect the x-axis. The correct answer is option a.
Discriminant Analysis
The discriminant D = b² – 4ac determines root nature for ax² + bx + c = 0. Here, a=1, b=-3, c=4, so D = (-3)² – 4(1)(4) = 9 – 16 = -7. Since D < 0, no real roots exist, confirming no x-axis intersection.
Option Breakdown
a. Does not intersect x-axis: True, as negative discriminant means the parabola stays above the x-axis (minimum value f(1.5) = 1.75 > 0).
b. Does not intersect y-axis: False, y-intercept at f(0) = 4.
c. Two real roots: False, requires D > 0.
d. One real, one imaginary root: False, quadratics have roots in conjugate pairs when complex.
Graph Behavior
Vertex at x = -b/(2a) = 1.5, where f(1.5) = 1.75 > 0. Y-intercept at (0,4) confirms y-axis crossing.
Exam Relevance
Negative discriminant (D = -7) means no real solutions to f(x) = 0, key for CSIR NET math sections.
| Discriminant Value | Nature of Roots | X-Axis Intersection |
|---|---|---|
| D > 0 | Two distinct real roots | Two intersections |
| D = 0 | One real root (repeated) | One intersection (tangent) |
| D < 0 (-7 in this case) | No real roots (complex conjugate pair) | No intersection |
