Discriminant and Nature of Roots MCQs Quiz | Class 10
This quiz is designed for Class: X students, covering Subject: Mathematics (Code 041), Unit: Unit II: Algebra, with a specific focus on the Topic: Discriminant and Nature of Roots MCQs Quiz | Class 10. The questions cover D=b^2−4ac; D>0, D=0, D<0 interpretation, emphasizing real roots. Test your understanding by attempting all questions, then click "Submit Quiz" to see your score and review answers, or "Download Answer PDF" for a detailed record.
Understanding Discriminant and Nature of Roots
The discriminant is a crucial part of a quadratic equation that helps us determine the nature of its roots without actually solving the equation. For a standard quadratic equation in the form ax^2 + bx + c = 0 (where a, b, and c are real numbers and a ≠ 0), the discriminant is denoted by D (or Δ) and is calculated using the formula:
The value of the discriminant tells us whether the roots are real or complex, and if real, whether they are distinct or equal.
Interpreting the Discriminant (D)
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Case 1: D > 0 (Discriminant is positive)
If the discriminant is greater than zero, the quadratic equation has two distinct real roots. This means the parabola representing the quadratic equation intersects the x-axis at two different points.
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Case 2: D = 0 (Discriminant is zero)
If the discriminant is equal to zero, the quadratic equation has two equal real roots. This is often referred to as having one real root with multiplicity two. Graphically, the parabola touches the x-axis at exactly one point.
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Case 3: D < 0 (Discriminant is negative)
If the discriminant is less than zero, the quadratic equation has no real roots. Instead, it has two distinct complex (or imaginary) roots. In this case, the parabola does not intersect the x-axis at all.
Summary Table: Nature of Roots
| Discriminant (D) | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots |
| D < 0 | No real roots (imaginary roots) |
Understanding the discriminant is essential for solving various problems related to quadratic equations, especially when you need to determine the feasibility of solutions in real-world contexts without going through the entire quadratic formula calculation.
Quick Revision Points
- The discriminant (D) is given by D = b^2 – 4ac for ax^2 + bx + c = 0.
- For real roots, D must be greater than or equal to zero (D >= 0).
- Distinct real roots occur when D > 0.
- Equal real roots occur when D = 0.
- No real roots (imaginary roots) occur when D < 0.
Practice Questions
Test your knowledge further with these practice questions:
- Find the discriminant of the quadratic equation 3x^2 – 5x + 7 = 0.
- Determine the nature of the roots for the equation 4x^2 – 2x + 1/4 = 0.
- For what value of ‘p’ does the quadratic equation px^2 + 6x + 1 = 0 have equal roots?
- Without solving, determine if the equation 2x^2 + x – 300 = 0 has real roots.
- If the roots of the equation 3x^2 + 2x + k = 0 are real, what condition must ‘k’ satisfy?
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