Nature of Roots of a Quadratic Equation
Nature of Roots of a Quadratic Equation describes whether its roots are real, equal, distinct, rational, irrational or non-real. The discriminant, D = b² − 4ac, gives the main test for a quadratic equation ax² + bx + c = 0, where a ≠ 0. Its value also helps form parameter-based conditions in aptitude problems.
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What Is the Nature of Roots of a Quadratic Equation?
For D > 0, the two roots are real and unequal. For D = 0, both roots are real and equal. For D < 0, the roots are a pair of non-real complex conjugates. When D > 0 and is a perfect square, rational coefficients give rational roots; when D > 0 but is not a perfect square, the roots are irrational.
Nature of Roots of a Quadratic Equation Video Lessons
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Quadratic Formula and Discriminant
Learn how to use the quadratic formula to solve quadratic equations and apply the discriminant to determine the nature and number of their roots.
Nature of Roots of a Quadratic Equation Formula & Tricks
Important Formulas
For ax² + bx + c = 0, calculate D using the coefficients a, b and c.
This formula gives both roots after the discriminant is found.
A zero discriminant means both roots have the same value.
These relations help determine the signs and possible values of the roots.
Quick Tricks
For deciding whether roots are real or non-real, the exact roots are not required. Compare b² with 4ac: b² > 4ac gives distinct real roots, b² = 4ac gives equal roots, and b² < 4ac gives non-real roots.
If D is positive and a, b and c are integers, a perfect-square D produces rational roots. A positive non-square D produces irrational roots.
Set D > 0, D = 0 or D < 0 according to the required type of roots, and then solve the resulting condition on the parameter.
Nature of Roots of a Quadratic Equation Concepts
Discriminant and Root Classification
For ax² + bx + c = 0 with a ≠ 0, D > 0 means two distinct real roots, D = 0 means two equal real roots, and D < 0 means two non-real complex conjugate roots. The number of roots counted with multiplicity remains two.
Rational and Irrational Real Roots
The quadratic formula contains √D. Thus, a perfect-square discriminant makes √D rational, while a positive non-square discriminant makes √D irrational. This classification applies after confirming that D > 0.
Conditions for Positive and Negative Roots
For roots α and β, α + β = −b/a and αβ = c/a. If D > 0 and αβ < 0, the roots have opposite signs. If D > 0, αβ > 0 and α + β > 0, both roots are positive. If D > 0, αβ > 0 and α + β < 0, both roots are negative.
Parameter-Based Conditions
For distinct real roots, use D > 0. For equal roots, use D = 0. For non-real roots, use D < 0. For example, x² + kx + 4 = 0 has distinct real roots when k² − 16 > 0, which gives k < −4 or k > 4.
Nature of Roots of a Quadratic Equation Data Table
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Classification Based on the Discriminant
Root type for ax² + bx + c = 0, where a ≠ 0
| Condition | Nature of roots | Additional property |
|---|---|---|
| D > 0 | Two real and distinct roots | Rational if D is a perfect square; otherwise irrational |
| D = 0 | Two real and equal roots | Common root is −b/2a |
| D < 0 | Two non-real complex conjugate roots | No real roots |
Note: Here D = b² − 4ac.
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Nature of Roots: Revision Points
Use the discriminant and root relations to classify and analyse quadratic roots.
- For ax² + bx + c = 0, a ≠ 0, D = b² − 4ac.
- D > 0 gives two real and unequal roots.
- D = 0 gives two equal real roots, each equal to −b/2a.
- D < 0 gives non-real complex conjugate roots.
- A positive perfect-square D gives rational roots when the coefficients are rational.
- A positive non-square D gives irrational roots when the coefficients are rational.
- α + β = −b/a and αβ = c/a.
- For parameter questions, use D > 0, D = 0 or D < 0 according to the required root type.
Nature of Roots of a Quadratic Equation FAQs
What is the formula for the nature of roots of a quadratic equation?
For ax² + bx + c = 0, calculate D = b² − 4ac. D > 0 gives distinct real roots, D = 0 gives equal real roots, and D < 0 gives non-real roots.
What is the nature of the roots of x² + 6x + 9 = 0?
D = 6² − 4(1)(9) = 36 − 36 = 0. Therefore, the roots are real and equal; both roots are −3.
When does a quadratic equation have irrational roots?
It has irrational roots when D is positive but not a perfect square, assuming the coefficients are rational. For x² − 4x + 1 = 0, D = 12, so its roots are irrational.
When does a quadratic equation have roots of opposite signs?
The roots have opposite signs when their product c/a is negative. In this case, D is automatically positive because b² − 4ac is greater than b².
For what values of k does x² + kx + 4 = 0 have non-real roots?
Require D < 0: k² − 16 < 0. Hence, −4 < k < 4.
Can a quadratic equation have one real root and one non-real root?
No. With real coefficients, a quadratic equation has either two real roots, two equal real roots, or a pair of non-real complex conjugate roots.