Discriminant of a Quadratic Equation

Q: What is the discriminant of a quadratic equation?

The discriminant is D = b² − 4ac, calculated from the coefficients of ax² + bx + c = 0. It determines the nature and number of roots.

Q: What does it mean when the discriminant is a perfect square?

When D is a positive perfect square and a, b, c are rational, the roots are rational numbers. The quadratic can be factorised into two linear factors with rational coefficients.

Q: Can the discriminant be negative?

Yes. When D < 0, the equation has no real roots because the square root of a negative number is not real. The roots are complex numbers, studied in higher classes.

Q: How do I find the value of k for equal roots?

Set D = 0. Substitute the values of a, b, c (which involve k) into D = b² − 4ac = 0, then solve the resulting equation for k.

Q: What is the discriminant of x² + 4x + 4 = 0?

a = 1, b = 4, c = 4. D = 4² − 4(1)(4) = 16 − 16 = 0. Since D = 0, the equation has two equal roots: x = −b/(2a) = −4/2 = −2.

Q: Is the discriminant the same as the roots of the equation?

No. The discriminant is D = b² − 4ac — a value that tells us about the nature of the roots. The actual roots are found using the quadratic formula x = (−b ± √D) / 2a.

Q: Why is the discriminant important in CBSE Class 10?

CBSE frequently asks questions about determining the nature of roots, finding unknown parameters for specific root conditions, and proving statements about quadratic equations. It is a high-scoring concept in board exams.

Q: Can a quadratic equation have exactly one root?

Every quadratic equation has exactly two roots (counting multiplicity). When D = 0, both roots are the same value — called a 'repeated' or 'double' root. It is more accurate to say two equal roots than one root.

Class 10Quadratic Equations

The discriminant determines the nature and number of roots of a quadratic equation without actually solving it. By computing a single value from the coefficients, students can immediately state whether the equation has two distinct real roots, one repeated root, or no real roots.

For a quadratic equation ax² + bx + c = 0, the discriminant is the expression D = b² − 4ac. It appears under the square root in the quadratic formula.

The discriminant also reveals how the parabola y = ax² + bx + c interacts with the x-axis — crossing at two points, touching at one, or not touching at all.

What is Discriminant?

Definition: The discriminant of a quadratic equation ax² + bx + c = 0 is the expression D = b² − 4ac.

It is denoted by D (or sometimes by the Greek letter Δ).

Key facts:

The discriminant is the portion of the quadratic formula under the square root sign.
It is a single numerical value derived from coefficients a, b, and c.
Its sign (positive, zero, or negative) fully determines the nature of the roots.
The discriminant is not a root of the equation.

Connection to Roots:

D > 0: Two distinct real roots.
D = 0: Two equal real roots (repeated root).
D < 0: No real roots (complex roots).

Additional insight when D > 0:

If D is a perfect square and a, b, c are rational → roots are rational → equation can be factorised.
If D is not a perfect square → roots are irrational (conjugate surds).

Graphical meaning: The discriminant tells us how many times the parabola crosses or touches the x-axis.

D > 0 → parabola crosses x-axis at two points
D = 0 → parabola touches x-axis at one point (vertex on x-axis)
D < 0 → parabola does not touch the x-axis

Discriminant of a Quadratic Equation Formula

Formula:

D = b² − 4ac

Where:

D = discriminant
a = coefficient of x²
b = coefficient of x
c = constant term

Connection to the Quadratic Formula:

x = (−b ± √D) / 2a, where D = b² − 4ac

Derivation and Proof

The discriminant arises during the derivation of the quadratic formula by completing the square.

Steps showing how D emerges:

Start with: ax² + bx + c = 0
Divide by a: x² + (b/a)x + c/a = 0
Rearrange: x² + (b/a)x = −c/a
Complete the square: (x + b/(2a))² = b²/(4a²) − c/a
Common denominator: (x + b/(2a))² = (b² − 4ac) / (4a²)

The expression b² − 4ac appears in the numerator. This is the discriminant.

How D determines the outcome:

Taking the square root: x + b/(2a) = ± √(b² − 4ac) / (2a)
If D > 0: √D is a positive real number → two distinct values from ±.
If D = 0: √D = 0 → the ± part vanishes → both roots become x = −b/(2a).
If D < 0: √D is not real → no real square root can be taken → no real roots.

The discriminant acts as the "switch" that determines which of the three outcomes occurs.

Types and Properties

Three Cases of the Discriminant:

Case	Value of D	Nature of Roots	Graph	Example
Case 1	D > 0	Two distinct real roots	Parabola crosses x-axis at two points	x² − 5x + 4 = 0 → D = 25 − 16 = 9 > 0
Case 2	D = 0	Two equal real roots	Parabola touches x-axis at one point	x² − 6x + 9 = 0 → D = 36 − 36 = 0
Case 3	D < 0	No real roots	Parabola does not touch x-axis	x² + 2x + 5 = 0 → D = 4 − 20 = −16 < 0

Sub-cases when D > 0:

Sub-condition	Nature of Roots	Example
D is a perfect square (with rational a, b, c)	Two distinct rational roots — factorisable	x² − 7x + 10 = 0 → D = 9 = 3²
D is not a perfect square	Two distinct irrational roots (conjugate surds)	x² − 3x + 1 = 0 → D = 5

Methods

Steps to Calculate and Interpret the Discriminant:

Write in standard form: ax² + bx + c = 0. Example: 3x² = 7x − 2 becomes 3x² − 7x + 2 = 0.
Identify a, b, c: Watch signs carefully. For 3x² − 7x + 2 = 0: a = 3, b = −7, c = 2.
Substitute into D = b² − 4ac: Remember b² means (b)², including the sign. If b = −7, then b² = (−7)² = 49 (not −49).
Interpret D:
- D > 0 → two distinct real roots
- D = 0 → two equal real roots (repeated root = −b/(2a))
- D < 0 → no real roots

Common Applications:

Finding the nature of roots without solving the equation.
Finding the value of an unknown parameter (k) for equal/distinct/no real roots.
Determining whether a quadratic can be factorised over the rationals.
Analysing the graph of a quadratic function.

Solved Examples

Example 1: Discriminant: Positive, Perfect Square

Problem: Find the discriminant of 2x² − 11x + 12 = 0 and state the nature of roots.

Solution:

Given:

a = 2, b = −11, c = 12

Using D = b² − 4ac:

D = (−11)² − 4(2)(12) = 121 − 96 = 25
D = 25 > 0 and 25 = 5² (perfect square)

Answer: D = 25. The equation has two distinct rational roots.

Example 2: Discriminant: Positive, Not a Perfect Square

Problem: Find the discriminant of x² + 5x + 3 = 0 and state the nature of roots.

Solution:

Given:

a = 1, b = 5, c = 3

Using D = b² − 4ac:

D = 5² − 4(1)(3) = 25 − 12 = 13
D = 13 > 0 but 13 is not a perfect square

Answer: D = 13. The equation has two distinct irrational roots (involving surds).

Example 3: Discriminant: Zero (Equal Roots)

Problem: Show that 4x² + 4x + 1 = 0 has equal roots and find the root.

Solution:

Given:

a = 4, b = 4, c = 1

Using D = b² − 4ac:

D = 4² − 4(4)(1) = 16 − 16 = 0
D = 0 → two equal roots

Finding the repeated root:

x = −b/(2a) = −4/(2 × 4) = −4/8 = −1/2

Answer: D = 0. Both roots equal −1/2.

Example 4: Discriminant: Negative (No Real Roots)

Problem: Find the discriminant of 3x² − 2x + 4 = 0 and comment on its roots.

Solution:

Given:

a = 3, b = −2, c = 4

Using D = b² − 4ac:

D = (−2)² − 4(3)(4) = 4 − 48 = −44
D = −44 < 0

Answer: D = −44. The equation has no real roots. The parabola y = 3x² − 2x + 4 does not intersect the x-axis.

Example 5: Finding k for Equal Roots

Problem: Find the value(s) of k for which x² + kx + 9 = 0 has equal roots.

Solution:

Given:

a = 1, b = k, c = 9

For equal roots, D = 0.

Steps:

D = k² − 4(1)(9) = k² − 36
Set D = 0: k² − 36 = 0
k² = 36
k = ± 6

Answer: k = 6 or k = −6. For both values, the equation has equal roots.

Example 6: Finding k for No Real Roots

Problem: Find the range of k for which 2x² + 5x + k = 0 has no real roots.

Solution:

Given:

a = 2, b = 5, c = k

For no real roots, D < 0.

Steps:

D = 5² − 4(2)(k) = 25 − 8k
Set D < 0: 25 − 8k < 0
8k > 25
k > 25/8 = 3.125

Answer: The equation has no real roots when k > 25/8 (i.e., k > 3.125).

Example 7: Finding p for Distinct Real Roots

Problem: For what values of p does px² + 8x + 2 = 0 have two distinct real roots?

Solution:

Given:

a = p, b = 8, c = 2
p ≠ 0 (otherwise not quadratic)

For distinct real roots, D > 0.

Steps:

D = 8² − 4(p)(2) = 64 − 8p
Set D > 0: 64 − 8p > 0
8p < 64 → p < 8
Combined with p ≠ 0: p < 8 and p ≠ 0

Answer: Two distinct real roots when p < 8 and p ≠ 0.

Example 8: Discriminant of a Rearranged Equation

Problem: Determine the nature of roots of (x − 3)(2x + 1) = 0.

Solution:

Expanding: 2x² + x − 6x − 3 = 0 → 2x² − 5x − 3 = 0

Given:

a = 2, b = −5, c = −3

Using D = b² − 4ac:

D = (−5)² − 4(2)(−3) = 25 + 24 = 49
D = 49 = 7² (perfect square)

From the factored form, roots are directly x = 3 and x = −1/2.

Answer: D = 49. Two distinct rational roots: x = 3 and x = −1/2.

Example 9: Proving Real Roots for a Parametric Equation

Problem: Prove that x² − 2(a + 1)x + a² = 0 always has real roots for any real value of a.

Solution:

Given:

A = 1, B = −2(a + 1), C = a²

Computing D:

D = [−2(a + 1)]² − 4(1)(a²)
D = 4(a + 1)² − 4a²
D = 4(a² + 2a + 1) − 4a²
D = 4a² + 8a + 4 − 4a² = 8a + 4 = 4(2a + 1)

For D ≥ 0: 2a + 1 ≥ 0, i.e., a ≥ −1/2.

Answer: D = 4(2a + 1). The equation has real roots when a ≥ −1/2. For a = −1/2, roots are equal.

Example 10: Discriminant with Parametric Coefficients

Problem: If 3x² + 2kx + 27 = 0 has real and equal roots, find k and the roots.

Solution:

Given:

a = 3, b = 2k, c = 27

For equal roots, D = 0.

Steps:

D = (2k)² − 4(3)(27) = 4k² − 324
Set D = 0: 4k² = 324 → k² = 81 → k = ± 9

Finding roots:

For k = 9: 3x² + 18x + 27 = 0 → x² + 6x + 9 = 0 → (x + 3)² = 0 → x = −3
For k = −9: 3x² − 18x + 27 = 0 → x² − 6x + 9 = 0 → (x − 3)² = 0 → x = 3

Answer: k = 9 (root x = −3) or k = −9 (root x = 3).

Real-World Applications

Quality of Solutions:

In engineering and science, calculating D before solving tells whether real solutions exist.
This saves time in analysis.

Graphical Analysis:

D determines the number of x-intercepts of the parabola y = ax² + bx + c.
Used in data fitting, curve sketching, and understanding quadratic models.

Optimisation Problems:

When a quadratic function models profit, trajectory, or area, D determines whether certain target values are achievable.

Geometry:

Finding whether a line intersects a circle: substitute the line into the circle equation and compute D.
D > 0 → secant (two intersection points)
D = 0 → tangent (one point)
D < 0 → no intersection

Number Theory:

D determines whether a quadratic has rational roots.
Used to decide whether a quadratic expression factors over the integers.

Key Points to Remember

The discriminant of ax² + bx + c = 0 is D = b² − 4ac.
D > 0: two distinct real roots. D = 0: two equal real roots. D < 0: no real roots.
If D is a perfect square and a, b, c are rational, the roots are rational and the equation can be factorised.
The discriminant appears under the square root in the quadratic formula: x = (−b ± √D) / 2a.
To find the value of a parameter for equal roots, set D = 0 and solve.
D tells the number of x-intercepts of the parabola: 2 (D > 0), 1 (D = 0), or 0 (D < 0).
Always write the equation in standard form before computing D.
b² always means the square of b including its sign: (−7)² = 49, not −49.
A common exam pattern: "Find the value of k for which the equation has equal/real/no real roots."
The discriminant is a value (positive, zero, or negative) — it is not a root of the equation.

Practice Problems

Find the discriminant and state the nature of roots: x² − 4x + 4 = 0
Find the discriminant: 3x² + 7x + 2 = 0. Are the roots rational or irrational?
Without solving, determine the nature of roots of 5x² − 6x + 3 = 0.
Find the values of k for which 2x² − kx + 3 = 0 has two equal roots.
For what values of m does x² + 2mx + m + 3 = 0 have (i) equal roots (ii) two distinct roots (iii) no real roots?
Prove that (k + 1)x² − 2(k − 1)x + 1 = 0 has real and equal roots when k = 0 or k = 3.

Frequently Asked Questions

Q1. What is the discriminant of a quadratic equation?

The discriminant is D = b² − 4ac, calculated from the coefficients of ax² + bx + c = 0. It determines the nature and number of roots.

Q2. How does the discriminant determine the nature of roots?

D > 0: two distinct real roots. D = 0: two equal real roots (repeated root). D < 0: no real roots (complex numbers).

Q3. What does it mean when the discriminant is a perfect square?

When D is a positive perfect square and a, b, c are rational, the roots are rational numbers. The quadratic can be factorised into two linear factors with rational coefficients.

Q4. Can the discriminant be negative?

Yes. When D < 0, the equation has no real roots because the square root of a negative number is not real. The roots are complex numbers, studied in higher classes.

Q5. How do I find the value of k for equal roots?

Set D = 0. Substitute the values of a, b, c (which involve k) into D = b² − 4ac = 0, then solve the resulting equation for k.

Q6. What is the discriminant of x² + 4x + 4 = 0?

a = 1, b = 4, c = 4. D = 4² − 4(1)(4) = 16 − 16 = 0. Since D = 0, the equation has two equal roots: x = −b/(2a) = −4/2 = −2.

Q7. Is the discriminant the same as the roots of the equation?

No. The discriminant is D = b² − 4ac — a value that tells us about the nature of the roots. The actual roots are found using the quadratic formula x = (−b ± √D) / 2a.

Q8. Why is the discriminant important in CBSE Class 10?

CBSE frequently asks questions about determining the nature of roots, finding unknown parameters for specific root conditions, and proving statements about quadratic equations. It is a high-scoring concept in board exams.

Q9. What is the graphical significance of the discriminant?

D > 0: parabola crosses the x-axis at two points. D = 0: parabola touches x-axis at one point (vertex). D < 0: parabola does not touch the x-axis.

Q10. Can a quadratic equation have exactly one root?

Every quadratic equation has exactly two roots (counting multiplicity). When D = 0, both roots are the same value — called a 'repeated' or 'double' root. It is more accurate to say two equal roots than one root.

Discriminant of a Quadratic Equation

What is Discriminant?

Discriminant of a Quadratic Equation Formula

Derivation and Proof

Types and Properties

Methods

Solved Examples

Example 1: Discriminant: Positive, Perfect Square

Example 2: Discriminant: Positive, Not a Perfect Square

Example 3: Discriminant: Zero (Equal Roots)

Example 4: Discriminant: Negative (No Real Roots)

Example 5: Finding k for Equal Roots

Example 6: Finding k for No Real Roots

Example 7: Finding p for Distinct Real Roots

Example 8: Discriminant of a Rearranged Equation

Example 9: Proving Real Roots for a Parametric Equation

Example 10: Discriminant with Parametric Coefficients

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is the discriminant of a quadratic equation?

Q2. How does the discriminant determine the nature of roots?

Q3. What does it mean when the discriminant is a perfect square?

Q4. Can the discriminant be negative?

Q5. How do I find the value of k for equal roots?

Q6. What is the discriminant of x² + 4x + 4 = 0?

Q7. Is the discriminant the same as the roots of the equation?

Q8. Why is the discriminant important in CBSE Class 10?

Q9. What is the graphical significance of the discriminant?

Q10. Can a quadratic equation have exactly one root?

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