Discriminant of a Quadratic Equation: Formula, Rules and Examples
Discriminant of a Quadratic Equation is the expression that determines the nature and number of its roots. For ax² + bx + c = 0, the discriminant is D = b² − 4ac. Its value shows whether the roots are real and distinct, real and equal, or non-real complex numbers.
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What Is the Discriminant of a Quadratic Equation?
If D > 0, the equation has two distinct real roots. If D = 0, it has two equal real roots. If D < 0, it has no real roots and its roots are a pair of complex conjugates. For example, in 2x² − 5x + 2 = 0, D = (−5)² − 4(2)(2) = 9, so the roots are real and distinct.
Discriminant 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.
Discriminant of a Quadratic Equation Formula & Tricks
Important Formulas
For ax² + bx + c = 0, a, b and c are the coefficients of x², x and the constant term.
This formula gives the roots after calculating D = b² − 4ac.
For roots α and β of ax² + bx + c = 0, these relations hold independently of the value of D.
When the discriminant is zero, both roots have the same value.
Quick Tricks
Write the equation in the standard form ax² + bx + c = 0 first. The sign of b must be retained, although b² removes its sign in the discriminant.
For questions asking only about the nature of roots, calculate or compare b² and 4ac. There is no need to find the roots.
If a quadratic must have equal roots, substitute its coefficients in b² − 4ac = 0 and solve for the parameter.
Discriminant of a Quadratic Equation Concepts
Nature of Roots from the Discriminant
For ax² + bx + c = 0: D > 0 gives two unequal real roots; D = 0 gives two equal real roots; and D < 0 gives two non-real complex conjugate roots. If D is a positive perfect square, the roots are rational; if D is positive but not a perfect square, the roots are irrational real numbers.
Finding the Discriminant from an Equation
The coefficient of x² is a, the coefficient of x is b, and the constant term is c. Terms missing from an equation have coefficient zero. For example, x² − 16 = 0 has a = 1, b = 0 and c = −16.
Equal and Repeated Roots
The root in this case is −b/(2a), and the equation can be written as a(x − r)² = 0, where r = −b/(2a). The repeated root is counted twice in the quadratic formula.
Discriminant in Parameter Problems
Use D = 0 for equal roots, D > 0 for two distinct real roots, and D < 0 for non-real roots. When a parameter occurs in a, b or c, simplify the resulting expression and apply the required sign condition.
Discriminant of a Quadratic Equation Data Table
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Root Conditions Based on D
The value of the discriminant classifies the roots of ax² + bx + c = 0.
| Condition on D | Nature of roots | Additional condition |
|---|---|---|
| D > 0 | Two distinct real roots | If D is a perfect square, roots are rational; otherwise, irrational |
| D = 0 | Two equal real roots | Repeated root is −b/(2a) |
| D < 0 | No real roots | Roots are complex conjugates |
Note: Here a, b and c are real numbers and a ≠ 0.
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Discriminant of a Quadratic Equation: Quick Revision
Use these rules to identify root types and solve coefficient-based questions.
- For ax² + bx + c = 0, D = b² − 4ac.
- D > 0 means two distinct real roots.
- D = 0 means two equal real roots.
- D < 0 means non-real complex conjugate roots.
- If D > 0 is a perfect square, the roots are rational.
- For equal roots, use D = 0 and repeated root = −b/(2a).
- Write the equation in standard form before identifying a, b and c.
- For parameter questions, apply D = 0, D > 0 or D < 0 according to the stated root condition.
Discriminant of a Quadratic Equation FAQs
What is the discriminant of a quadratic equation?
For ax² + bx + c = 0, the discriminant is D = b² − 4ac. It determines the nature of the roots.
What happens when the discriminant is zero?
When D = 0, the quadratic has two equal real roots. The repeated root is −b/(2a).
What does a negative discriminant indicate?
If D < 0, the quadratic has no real roots. Its roots are a pair of non-real complex conjugates.
When are the roots of a quadratic rational?
The roots are rational when D is positive and a perfect square, provided the coefficients are rational. For example, D = 9 gives rational roots.
Find the discriminant of 5x² + 2x − 3 = 0.
Here a = 5, b = 2 and c = −3. D = 2² − 4(5)(−3) = 4 + 60 = 64.
For which values of k does x² + kx + 25 = 0 have equal roots?
Set D = 0: k² − 4(1)(25) = 0, so k² = 100. Therefore, k = 10 or k = −10.