Nature of Roots of Quadratic Equation

Q: What is the difference between real roots and no real roots?

Real roots are values on the number line that satisfy the equation. "No real roots" means no number-line value satisfies it. The solutions exist as complex numbers (involving √(−1)), studied in higher classes.

Q: What does it mean when roots are 'equal'?

When D = 0, the formula gives x = −b/(2a) for both roots. Both solutions are the same number — a "double root" or "repeated root". Graphically, the parabola touches the x-axis at its vertex.

Q: Can a quadratic equation with real coefficients have one real and one imaginary root?

No. With real coefficients, roots always come in matching pairs: either both are real (distinct or equal), or both are complex conjugates (a + bi and a − bi). One real and one imaginary is not possible.

Q: Why do irrational roots come in conjugate pairs?

From the quadratic formula, roots are (−b + √D)/(2a) and (−b − √D)/(2a). When D is not a perfect square, √D is irrational, producing roots of the form p + √q and p − √q. These conjugate surds always occur together.

Q: What type of questions on nature of roots come in CBSE board exams?

Common patterns: finding the discriminant and stating the nature; finding k for equal/distinct/no real roots; proving an equation always has real roots; word problems asking whether a situation is possible (justified using D).

Q: If the discriminant of an equation is 0, does the equation have only one solution?

The equation has two solutions, but both are the same value — a "repeated root of multiplicity 2". Some textbooks say "only one root" for simplicity, but technically it has two equal roots.

Class 10Quadratic Equations

The nature of roots of a quadratic equation tells us the type of solutions an equation will yield without actually solving it.

This analysis uses the discriminant, D = b² − 4ac, which classifies the roots of any quadratic equation ax² + bx + c = 0.

The discriminant determines:

Whether roots are real or imaginary
Whether they are equal or distinct
Whether they are rational or irrational

This concept appears consistently in CBSE board exams and is used in graphical analysis, optimisation problems, and feasibility checks in physics and engineering.

What is Nature of Roots?

Definition: The nature of roots of a quadratic equation ax² + bx + c = 0 (where a ≠ 0) is the characterisation of its solutions based on the value of the discriminant D = b² − 4ac.

The quadratic formula is:

x = (−b ± √D) / 2a

Since the formula involves √D, the value of D controls the type of roots:

D > 0: √D is a positive real number → two different real roots
D = 0: √D = 0 → both roots equal −b/(2a)
D < 0: √D is not a real number → no real roots

The nature of roots is determined without solving the equation. By computing a single number (the discriminant), we gain complete qualitative information about the roots.

In the NCERT Class 10 syllabus, students handle three cases:

D > 0: Two distinct real roots
D = 0: Two equal real roots
D < 0: No real roots

Nature of Roots of Quadratic Equation Formula

Discriminant formula:

D = b² − 4ac

Classification of roots:

Condition	Nature of Roots
D > 0	Two distinct real roots: x = (−b + √D)/2a and x = (−b − √D)/2a
D = 0	Two equal real roots: x = −b/(2a) (repeated root)
D < 0	No real roots (roots are complex conjugates)

Further classification when D > 0:

If D is a perfect square → roots are rational (equation can be factorised)
If D is not a perfect square → roots are irrational (conjugate surds: p ± √q)

Derivation and Proof

Derivation of the nature of roots from the quadratic formula:

Step 1: The roots of ax² + bx + c = 0 are:

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

Step 2 — Case D > 0:

√D is a positive real number.
The ± sign produces two different values: x₁ = (−b + √D)/2a and x₂ = (−b − √D)/2a.
Since √D ≠ 0, x₁ ≠ x₂. Roots are two distinct real numbers.

Step 3 — Sub-case: D is a perfect square:

If D = k² for some rational k, then √D = k (rational).
Both roots are rational. The quadratic can be factorised over rationals.

Step 4 — Sub-case: D is not a perfect square:

√D is irrational. Roots are p + q√D and p − q√D (conjugate surds).
Irrational roots always occur in conjugate pairs.

Step 5 — Case D = 0:

x₁ = (−b + 0)/2a = −b/(2a)
x₂ = (−b − 0)/2a = −b/(2a)
Both roots are the same: repeated or double root.

Step 6 — Case D < 0:

√D = √(negative number), which is not a real number.
No real roots exist. Roots are complex numbers (studied in Class 11).

Step 7 — Graphical confirmation:

Roots = x-intercepts of the parabola y = ax² + bx + c.
D > 0: parabola crosses x-axis at two points.
D = 0: parabola touches x-axis at vertex (one point).
D < 0: parabola does not touch x-axis.

Types and Properties

Complete classification of roots:

Discriminant (D)	Nature of Roots	Graph of Parabola	Factorisable?
D > 0, perfect square	Two distinct rational roots	Crosses x-axis at 2 points	Yes
D > 0, not perfect square	Two distinct irrational roots (conjugate surds)	Crosses x-axis at 2 points	No (over rationals)
D = 0	Two equal (repeated) roots	Touches x-axis at 1 point	Yes (perfect square)
D < 0	No real roots	Does not touch x-axis	No

Important properties:

Irrational roots always occur in conjugate pairs: if (3 + √5) is a root, then (3 − √5) is also a root.
Complex roots also occur in conjugate pairs (Class 11).
Equal roots mean the quadratic is a perfect square: ax² + bx + c = a(x − r)².

Methods

Method for determining nature of roots:

Write the equation in standard form: ax² + bx + c = 0.
Identify a, b, and c carefully (note signs).
Compute D = b² − 4ac.
Classify based on the value of D.

Method for "find k" questions:

If the question says...	Set this condition
"has equal roots"	D = 0
"has real roots"	D ≥ 0
"has real and equal roots"	D = 0
"has two distinct real roots"	D > 0
"has no real roots"	D < 0
"has rational roots"	D > 0 and D is a perfect square

Method for graphical questions:

The number of x-intercepts of y = ax² + bx + c equals the number of distinct real roots:

D > 0 → 2 intersection points
D = 0 → 1 point (tangent to x-axis)
D < 0 → 0 points

Solved Examples

Example 1: Determining Nature with All Positive Coefficients

Problem: Without solving, find the nature of roots of 2x² + 9x + 4 = 0.

Solution:

Given: a = 2, b = 9, c = 4

D = 9² − 4(2)(4) = 81 − 32 = 49
D = 49 > 0. Also, 49 = 7² is a perfect square.

Answer: Two distinct real and rational roots. The equation can be factorised.

Example 2: Nature of Roots with Negative Coefficients

Problem: Determine the nature of roots of x² − 4x − 1 = 0.

Solution:

Given: a = 1, b = −4, c = −1

D = (−4)² − 4(1)(−1) = 16 + 4 = 20
D = 20 > 0, but 20 is not a perfect square (√20 = 2√5).

Answer: Two distinct real but irrational roots. The roots are conjugate surds: 2 + √5 and 2 − √5.

Example 3: Proving Equal Roots

Problem: Show that (a − b)x² + (b − c)x + (c − a) = 0 has equal roots when 2a = b + c.

Solution:

Given: A = (a − b), B = (b − c), C = (c − a). Also 2a = b + c, so b = 2a − c.

Substituting b = 2a − c:

A = (a − 2a + c) = (c − a)
B = (2a − c − c) = 2(a − c)
C = (c − a)

Computing D:

D = [2(a − c)]² − 4(c − a)(c − a)
D = 4(a − c)² − 4(c − a)²
Since (a − c)² = (c − a)²: D = 4(c − a)² − 4(c − a)² = 0

Answer: D = 0, so the equation has equal roots when 2a = b + c (a, b, c are in AP).

Example 4: Finding k for Equation with Equal Roots

Problem: Find k for which 4x² − 2(k + 1)x + (k + 1) = 0 has two equal roots.

Solution:

Given: a = 4, b = −2(k + 1), c = (k + 1)

For equal roots: D = 0

D = [−2(k + 1)]² − 4(4)(k + 1)
D = 4(k + 1)² − 16(k + 1)
D = 4(k + 1)[(k + 1) − 4] = 4(k + 1)(k − 3) = 0

Solutions: k + 1 = 0 → k = −1, or k − 3 = 0 → k = 3.

Verification:

k = −1: equation becomes 4x² = 0, so x = 0 (equal roots ✓)
k = 3: equation becomes 4x² − 8x + 4 = 0, i.e., (2x − 2)² = 0, so x = 1 (equal roots ✓)

Answer: k = −1 or k = 3.

Example 5: Finding Range of k for Real Roots

Problem: Find the values of k for which x² + k(2x + k − 1) + 2 = 0 has real roots.

Solution:

Expand: x² + 2kx + k² − k + 2 = 0

Given: a = 1, b = 2k, c = k² − k + 2

For real roots: D ≥ 0

D = (2k)² − 4(1)(k² − k + 2)
D = 4k² − 4k² + 4k − 8 = 4k − 8
4k − 8 ≥ 0 → 4k ≥ 8 → k ≥ 2

Answer: The equation has real roots when k ≥ 2.

Example 6: Checking if a Line Touches a Curve

Problem: How many times does the line y = 2x + 3 intersect the parabola y = x² + x + 1?

Solution:

At intersection: 2x + 3 = x² + x + 1

Rearranging: x² − x − 2 = 0

D = (−1)² − 4(1)(−2) = 1 + 8 = 9
D = 9 > 0 → two distinct real roots

Answer: The line intersects the parabola at two distinct points.

Example 7: Nature of Roots — Equation from Word Problem

Problem: Is it possible to design a rectangular park of perimeter 40 m and area 400 m²?

Solution:

Given:

Let length = x m. Then breadth = (20 − x) m (half perimeter = 20).
Area: x(20 − x) = 400

Forming equation: x² − 20x + 400 = 0

D = (−20)² − 4(1)(400) = 400 − 1600 = −1200
D < 0 → no real roots

Answer: Not possible. A rectangular park with perimeter 40 m and area 400 m² cannot exist.

Example 8: Both Equal and No Real Roots Conditions

Problem: For (k − 12)x² + 2(k − 12)x + 2 = 0, find k for (a) equal roots and (b) no real roots.

Solution:

Given: a = (k − 12), b = 2(k − 12), c = 2. Note: k ≠ 12.

D = [2(k − 12)]² − 4(k − 12)(2)
D = 4(k − 12)² − 8(k − 12)
D = 4(k − 12)[(k − 12) − 2] = 4(k − 12)(k − 14)

(a) Equal roots: D = 0 → (k − 12)(k − 14) = 0 → k = 12 or k = 14. Since k ≠ 12, k = 14.

(b) No real roots: D < 0 → (k − 12)(k − 14) < 0 → 12 < k < 14.

Answer: (a) k = 14; (b) 12 < k < 14.

Example 9: Proving Roots Are Always Real

Problem: Prove that (a² + b²)x² + 2(ac + bd)x + (c² + d²) = 0 has real roots only when ad = bc.

Solution:

D = [2(ac + bd)]² − 4(a² + b²)(c² + d²)
D = 4[(a²c² + 2abcd + b²d²) − (a²c² + a²d² + b²c² + b²d²)]
D = 4[2abcd − a²d² − b²c²] = −4[a²d² − 2abcd + b²c²]
D = −4(ad − bc)²
Since (ad − bc)² ≥ 0, D = −4(ad − bc)² ≤ 0.
D = 0 only when ad − bc = 0, i.e., ad = bc.

Answer: The roots are real (D ≥ 0) only when ad = bc. In that case, D = 0 and roots are equal.

Example 10: Using Nature of Roots in a Practical Context

Problem: A ball is thrown upward at 20 m/s from a 25 m platform. Height: h = 25 + 20t − 5t². Can it reach 50 m?

Solution:

Setting h = 50:

50 = 25 + 20t − 5t²
5t² − 20t + 25 = 0 → t² − 4t + 5 = 0

Discriminant:

D = (−4)² − 4(1)(5) = 16 − 20 = −4
D < 0 → no real roots

No real value of t gives h = 50.

Maximum height: At t = −b/(2a) = 4/2 = 2 seconds, h = 25 + 40 − 20 = 45 m.

Answer: No, the ball cannot reach 50 m. Maximum height is 45 m.

Real-World Applications

Applications of the nature of roots:

Feasibility Analysis: Before solving a real-world quadratic, check D. If D < 0, the physical situation is impossible (e.g., a room with given dimensions cannot exist).
Geometry — Tangency: When a line y = mx + c intersects a circle x² + y² = r², the resulting quadratic has D > 0 (secant, 2 points), D = 0 (tangent, 1 point), or D < 0 (no intersection).
Physics — Projectile Motion: The discriminant determines whether a projectile can reach a target height. D < 0 means the height is unachievable.
Economics — Break-even Analysis: For profit function P(x) = −ax² + bx − c, if D < 0, the company never breaks even.
Optimisation: The vertex at x = −b/(2a) gives the maximum or minimum. The discriminant indicates whether the function crosses zero.

Key Points to Remember

The nature of roots is determined by D = b² − 4ac without solving the equation.
D > 0 → two distinct real roots; D = 0 → two equal real roots; D < 0 → no real roots.
When D is a positive perfect square with rational coefficients, roots are rational and the equation can be factorised.
Irrational roots always occur in conjugate surd pairs (e.g., 3 + √2 and 3 − √2).
For "find k" questions: set D = 0 (equal), D > 0 (distinct), D < 0 (no real), or D ≥ 0 (real roots).
Graphical interpretation: D determines the number of x-intercepts of y = ax² + bx + c.
"Has real roots" means D ≥ 0 (includes equal roots). Read questions carefully.
Always verify the equation remains quadratic (a ≠ 0) after substituting k.
Common trap: "has real roots" requires D ≥ 0, not D > 0.
The discriminant is always a real number — it can be positive, zero, or negative.

Practice Problems

Find the nature of roots of 3x² - 4√3x + 4 = 0.
Without solving, determine if x² + 2x + 9 = 0 has real roots.
Find the value of k for which 2x² - 3x + k = 0 has equal roots.
For what values of p does (p + 1)x² - 6(p + 1)x + 3(p + 9) = 0 have equal roots? (p ≠ -1)
Determine the nature of roots of x² - 2x + 3 = 0. What does this tell us about the graph of y = x² - 2x + 3?
Find the values of k for which kx² + 2kx + 6 = 0 has (a) equal roots (b) no real roots.
If the roots of the equation (c² - ab)x² - 2(a² - bc)x + (b² - ac) = 0 are real and equal, show that either a = 0 or a³ + b³ + c³ = 3abc.
A ball is thrown such that its height h = 40t - 5t². Can the ball reach a height of 100 m? Justify your answer.

Frequently Asked Questions

Q1. What is meant by the nature of roots of a quadratic equation?

The nature of roots is the classification of solutions based on the discriminant D = b2 − 4ac. It tells whether roots are real or imaginary, distinct or equal, and rational or irrational.

Q2. How do you find the nature of roots without solving the equation?

Calculate D = b2 − 4ac. If D > 0, roots are distinct and real. If D = 0, roots are real and equal. If D < 0, no real roots exist. This determines the nature without finding actual root values.

Q3. What is the difference between real roots and no real roots?

Real roots are values on the number line that satisfy the equation. "No real roots" means no number-line value satisfies it. The solutions exist as complex numbers (involving √(−1)), studied in higher classes.

Q4. What does it mean when roots are 'equal'?

When D = 0, the formula gives x = −b/(2a) for both roots. Both solutions are the same number — a "double root" or "repeated root". Graphically, the parabola touches the x-axis at its vertex.

Q5. Can a quadratic equation with real coefficients have one real and one imaginary root?

No. With real coefficients, roots always come in matching pairs: either both are real (distinct or equal), or both are complex conjugates (a + bi and a − bi). One real and one imaginary is not possible.

Q6. What is the difference between D > 0 and D ≥ 0?

D > 0 means two distinct (different) real roots. D ≥ 0 means real roots that could be either distinct (D > 0) or equal (D = 0). When a question asks for "real roots", use D ≥ 0.

Q7. How are the nature of roots and the graph of a parabola related?

The roots of ax2 + bx + c = 0 are the x-intercepts of y = ax2 + bx + c. D > 0: parabola crosses x-axis at two points. D = 0: touches at one point. D < 0: does not touch the x-axis.

Q8. Why do irrational roots come in conjugate pairs?

From the quadratic formula, roots are (−b + √D)/(2a) and (−b − √D)/(2a). When D is not a perfect square, √D is irrational, producing roots of the form p + √q and p − √q. These conjugate surds always occur together.

Q9. What type of questions on nature of roots come in CBSE board exams?

Common patterns: finding the discriminant and stating the nature; finding k for equal/distinct/no real roots; proving an equation always has real roots; word problems asking whether a situation is possible (justified using D).

Q10. If the discriminant of an equation is 0, does the equation have only one solution?

The equation has two solutions, but both are the same value — a "repeated root of multiplicity 2". Some textbooks say "only one root" for simplicity, but technically it has two equal roots.

Nature of Roots of Quadratic Equation

What is Nature of Roots?

Nature of Roots of Quadratic Equation Formula

Derivation and Proof

Types and Properties

Methods

Solved Examples

Example 1: Determining Nature with All Positive Coefficients

Example 2: Nature of Roots with Negative Coefficients

Example 3: Proving Equal Roots

Example 4: Finding k for Equation with Equal Roots

Example 5: Finding Range of k for Real Roots

Example 6: Checking if a Line Touches a Curve

Example 7: Nature of Roots — Equation from Word Problem

Example 8: Both Equal and No Real Roots Conditions

Example 9: Proving Roots Are Always Real

Example 10: Using Nature of Roots in a Practical Context

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is meant by the nature of roots of a quadratic equation?

Q2. How do you find the nature of roots without solving the equation?

Q3. What is the difference between real roots and no real roots?

Q4. What does it mean when roots are 'equal'?

Q5. Can a quadratic equation with real coefficients have one real and one imaginary root?

Q6. What is the difference between D > 0 and D ≥ 0?

Q7. How are the nature of roots and the graph of a parabola related?

Q8. Why do irrational roots come in conjugate pairs?

Q9. What type of questions on nature of roots come in CBSE board exams?

Q10. If the discriminant of an equation is 0, does the equation have only one solution?

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