Quadratic Equations — Complete Lecture

Dive deep into the world of Quadratic Equations — understand their structure, methods of solving, and real-world applications. Perfect for JEE, NEET, and other competitive exams.

What You'll Learn:

In this lecture, you will gain a complete understanding of Quadratic Equations — one of the most fundamental and high-scoring topics in mathematics. We’ll begin by exploring the definition and standard form of quadratic equations and gradually build up to solving them using different methods such as factorization, completing the square, and the quadratic formula. You’ll also discover how to analyze the nature of roots using the discriminant (Δ), understand the relationship between coefficients and roots, and interpret the graphical behavior of parabolas. Each concept will be explained step-by-step with examples, visual demonstrations, and intuitive reasoning to ensure deep conceptual clarity. By the end of this lecture, you’ll be able to solve any quadratic problem confidently — from basic theory-based questions to complex problems asked in JEE, NEET, and other competitive exams.

🎥 Quadratic Equations – Full Lecture Video

Watch this complete lecture to understand every aspect of quadratic equations in detail — from their basic definition to advanced problem-solving techniques. The session includes conceptual breakdowns, real-time problem solving, and smart tricks for competitive exams.

Quadratic Equations — Complete Concept Guide

Click on any topic below to explore its explanation, formulas, and examples.

Introduction & Standard Form of a Quadratic Equation

A Quadratic Equation is an equation in which the highest power of the variable (usually x) is 2. It can be written in the standard form as:

ax² + bx + c = 0

Here:

  • a, b, and c are real numbers (called coefficients).
  • a ≠ 0 — because if a = 0, the equation becomes linear (not quadratic).

The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

When we graph a quadratic equation, it forms a parabola — a curved shape that can open upwards or downwards depending on the sign of a.

Tip for Exams: Remember — every quadratic equation can be solved using different methods such as factorization, completing the square, or the quadratic formula. Mastering all methods is essential for JEE, NEET, and other competitive exams.

Forms of Quadratic Equations

A quadratic equation can be expressed in different forms depending on how we want to analyze or graph it. Each form highlights different properties like roots, vertex, or coefficients.

1️⃣ General (or Standard) Form

Equation: ax² + bx + c = 0

This is the most common form, where a, b, and c are constants and a ≠ 0. It is mainly used for applying solving methods like the quadratic formula or factorization.

2️⃣ Factored Form

Equation: a(x - r₁)(x - r₂) = 0

This form directly shows the roots (r₁ and r₂) of the quadratic equation. It is obtained by factoring the equation or using known roots. Useful for quickly identifying where the graph crosses the x-axis.

3️⃣ Vertex Form

Equation: a(x - h)² + k = 0

This form shows the vertex of the parabola at point (h, k). It’s very helpful for graphing and understanding the parabola’s maximum or minimum value.

Quick Tip: - Use General Form for solving.
- Use Factored Form to find roots.
- Use Vertex Form to study graph shape and turning points.

Methods of Solving Quadratic Equations

A quadratic equation can be solved in several ways. The goal is to find the values of x that satisfy the equation ax² + bx + c = 0. Below are the four main solving methods — each useful in different situations.

1️⃣ Factorization Method

Split the middle term (bx) into two terms so that the product of their coefficients equals a × c. Then, group terms and factorize.
Example: x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2, -3.
Best used when coefficients are simple integers.

2️⃣ Completing the Square Method

Rearrange the equation to form a perfect square trinomial on one side. Then take square roots on both sides to solve for x.
Example: x² + 6x + 5 = 0 → (x + 3)² = 4 → x = -3 ± 2.
Useful for deriving the quadratic formula and understanding graph symmetry.

3️⃣ Quadratic Formula Method

This is the most general and powerful method. For any quadratic equation ax² + bx + c = 0, the roots are given by:

x = (-b ± √(b² - 4ac)) / 2a

Discriminant (D) = b² - 4ac helps determine the nature of roots:

  • D > 0 → Two distinct real roots
  • D = 0 → Two equal real roots
  • D < 0 → No real roots (complex roots)

Used in almost all competitive exams and engineering problems.

4️⃣ Graphical Method

The quadratic equation ax² + bx + c = 0 is represented by a parabola. The points where the parabola cuts the x-axis are the real roots of the equation.
Helps visualize the nature and number of roots.

Quick Tip for Exams: - Try factorization first for simple numbers.
- Use formula or completing square for tricky ones.
- Use graph to visualize or check your results.

Discriminant & Nature of Roots

The Discriminant, denoted by D, is the key quantity that helps us determine the nature of roots of a quadratic equation ax² + bx + c = 0.

D = b² − 4ac

The value of D tells us how many and what type of roots the quadratic equation has:

Condition on D Nature of Roots Type
D > 0 Two distinct real roots The parabola cuts the x-axis at two different points.
D = 0 Two equal real roots The parabola touches the x-axis at one point (double root).
D < 0 Complex (non-real) conjugate roots The parabola does not cut the x-axis.
Remember: - Discriminant helps us predict the graph’s intersection with the x-axis without actually solving the equation.
- For JEE and NEET, questions often test how changing a, b, or c affects the value of D.

Relations Between Roots & Coefficients

For a quadratic equation ax² + bx + c = 0, let the roots be α and β. These roots are connected to the coefficients a, b, and c through simple and powerful relationships.

a x² + b x + c = 0

The relations are:

  • Sum of roots (α + β) = −b / a
  • Product of roots (αβ) = c / a
Proof (Conceptual Understanding)

If the roots of the equation are α and β, then we can write it as:
a(x − α)(x − β) = 0

Expanding this gives:

a(x² − (α + β)x + αβ) = 0

Comparing with ax² + bx + c = 0, we get:

  • −(α + β) = b/a → α + β = −b/a
  • αβ = c/a
Quick Tip for Exams: - These relations are very useful in forming new equations when roots are modified (e.g., doubled, squared, or shifted).
- Many JEE/NEET questions are based on using these formulas instead of solving the whole quadratic.

Vieta’s Theorem and Proof

Vieta’s Theorem (also called Vieta’s Formula) connects the coefficients of a polynomial with the sum and product of its roots. It allows us to find relationships between roots without solving the equation directly — very useful in algebra and competitive exams.

For a Quadratic Equation:

a x² + b x + c = 0

Let the roots be α and β. Then, according to Vieta’s Theorem:

  • Sum of roots: α + β = −b / a
  • Product of roots: αβ = c / a
Proof of Vieta’s Theorem (for Quadratic Equations)

If the roots of the quadratic equation are α and β, the equation can be written as:

a(x − α)(x − β) = 0

Expanding this gives:

a(x² − (α + β)x + αβ) = 0

Comparing with the standard form ax² + bx + c = 0, we get:

  • −(α + β) = b/a ⇒ α + β = −b/a
  • αβ = c/a
Generalized Vieta’s Theorem (for Higher Degree Polynomials)

For a polynomial of degree n:

a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² + … + aₙ = 0

with roots r₁, r₂, r₃, …, rₙ, we have:

  • r₁ + r₂ + r₃ + … + rₙ = −a₁ / a₀
  • r₁r₂ + r₁r₃ + … = a₂ / a₀
  • r₁r₂r₃ + … = −a₃ / a₀
  • ⋯
  • r₁r₂…rₙ = (−1)ⁿ × aₙ / a₀
Quick Tip for Exams: - Vieta’s relations are extremely useful in forming new equations, finding missing coefficients, and solving root-based problems.
- In JEE & NEET, many algebra questions are based on applying these relations logically instead of solving the full quadratic.

Formation of Quadratic Equations

The process of forming a quadratic equation means creating an equation when its roots (α and β) are known. This is a reverse application of Vieta’s Theorem and is very useful in algebra and competitive exams.

If roots are α and β, then the equation is:
a(x − α)(x − β) = 0

Expanding this gives the standard quadratic form:

a(x² − (α + β)x + αβ) = 0

or equivalently, x² − (α + β)x + αβ = 0 when a = 1.

Example 1:

Form a quadratic equation whose roots are 2 and 3.

Using the formula: (x − 2)(x − 3) = 0 ⇒ x² − 5x + 6 = 0

Hence, the required quadratic equation is x² − 5x + 6 = 0.

Example 2:

Form a quadratic equation whose roots are 2p and 1/p.

Sum of roots = 2p + 1/p Product of roots = 2
So, equation ⇒ x² − (2p + 1/p)x + 2 = 0

Quick Tip for Exams: - Always find the sum (α + β) and product (αβ) of the given roots first.
- Substitute them in x² − (α + β)x + αβ = 0.
- For JEE/NEET-level questions, roots are often in special forms (like 1/α, α², 2α + 3β, etc.) — practice forming equations in those cases too.

Graph of a Quadratic Function

The graph of a quadratic function of the form y = ax² + bx + c is called a parabola. The shape and position of the parabola depend on the coefficients a, b, and c.

y = ax² + bx + c
1️⃣ Direction of the Parabola
  • If a > 0 → Parabola opens upward (U-shaped).
  • If a < 0 → Parabola opens downward (inverted U-shaped).

The coefficient ‘a’ controls how “wide” or “narrow” the parabola appears.

2️⃣ Vertex of the Parabola

The vertex is the turning point of the parabola — it represents the maximum or minimum value of the function.

Vertex (h, k) = (−b / 2a , (4ac − b²) / 4a)

  • If a > 0 → vertex is the minimum point.
  • If a < 0 → vertex is the maximum point.
3️⃣ Axis of Symmetry

The parabola is symmetric about the vertical line passing through its vertex. Equation of the axis of symmetry:

x = −b / 2a

4️⃣ Intercepts
  • Y-intercept: The point where the graph cuts the y-axis (x = 0) → y = c
  • X-intercepts: The points where y = 0 (found by solving ax² + bx + c = 0)
Quick Tip for Exams: - The vertex form y = a(x − h)² + k is best for graphing. - D = b² − 4ac determines how many x-intercepts (real roots) exist. - Graph-based questions often ask for the vertex, direction, and number of real roots.

Sign of Expression & Intervals

The sign of a quadratic expression (ax² + bx + c) tells us where the expression is positive or negative on the x-axis. This concept is essential for solving inequalities and optimization problems in exams.

Expression: ax² + bx + c
1️⃣ Case 1 — When a > 0 (Parabola Opens Upward)
  • The expression ax² + bx + c > 0 (positive) when x lies outside the roots.
  • The expression ax² + bx + c < 0 (negative) when x lies between the roots.

Sign pattern → (+) (α) (−) (β) (+)

2️⃣ Case 2 — When a < 0 (Parabola Opens Downward)
  • The expression ax² + bx + c > 0 (positive) when x lies between the roots.
  • The expression ax² + bx + c < 0 (negative) when x lies outside the roots.

Sign pattern → (−) (α) (+) (β) (−)

3️⃣ When the Roots are Not Real (D < 0)

Since the parabola does not cut the x-axis, the expression never changes sign. The sign of a decides the overall sign:

  • If a > 0 → expression is always positive.
  • If a < 0 → expression is always negative.
4️⃣ Interval Representation

Let the roots be α and β (α < β). Then:

  • If a > 0 → Expression > 0 for (−∞, α) ∪ (β, ∞)
  • If a > 0 → Expression < 0 for (α, β)
  • If a < 0 → Expression < 0 for (−∞, α) ∪ (β, ∞)
  • If a < 0 → Expression > 0 for (α, β)
Quick Tip for Exams: - Always check D = b² − 4ac first — it tells whether roots are real or not.
- Draw a quick parabola sketch to visualize signs easily.
- This topic is frequently used in inequality and graph-based JEE/NEET questions.

Maximum & Minimum Values

The maximum or minimum value of a quadratic function y = ax² + bx + c occurs at its vertex.

x-coordinate of Vertex → x = −b / (2a)

Substitute this x-value in the equation y = ax² + bx + c to get the extreme (maximum or minimum) value of y:

Extreme value (yₑ) = c − (b² / 4a)
1️⃣ Nature of the Vertex
  • If a > 0 → Parabola opens upward, and the vertex gives the minimum value.
  • If a < 0 → Parabola opens downward, and the vertex gives the maximum value.
2️⃣ Formula Summary
Coefficient ‘a’ Type of Extremum x-coordinate y-coordinate (Value)
a > 0 Minimum −b / 2a c − (b² / 4a)
a < 0 Maximum −b / 2a c − (b² / 4a)
3️⃣ Quick Example

For y = 2x² − 8x + 5:

  • a = 2, b = −8, c = 5
  • x = −(−8) / (2×2) = 2
  • y = 2(2)² − 8(2) + 5 = −3

✅ Minimum value = −3 (at x = 2)

Exam Tip: - The vertex form of a quadratic function: y = a(x − h)² + k, where vertex = (h, k). - The value of y at vertex (k) is the max/min depending on the sign of a. - Commonly asked in JEE, NEET, and Board exams to find min/max value or point.

Parametric & Condition-Based Equations

In many problems, quadratic equations involve one or more parameters (like p, k, m, or λ). These parameters affect the nature of roots, and we must find their possible values that satisfy certain conditions — such as having real roots, equal roots, or roots in a specific ratio or interval.

1️⃣ Condition for Real, Equal, and Imaginary Roots

Given a quadratic equation ax² + bx + c = 0 (where coefficients may include parameters):

  • D = b² − 4ac > 0 → Two distinct real roots
  • D = b² − 4ac = 0 → Two equal real roots
  • D = b² − 4ac < 0 → Complex (non-real) roots

These conditions are often used to find range or specific values of parameters.

2️⃣ Example: Find Range of Parameter

For equation x² + (k − 2)x + k = 0, find the condition for real roots.

Solution:

  • D = (k − 2)² − 4(1)(k) ≥ 0
  • ⇒ k² − 4k + 4 − 4k ≥ 0
  • ⇒ k² − 8k + 4 ≥ 0
  • ⇒ k ≤ 4 − 2√3 or k ≥ 4 + 2√3

✅ Hence, real roots exist for k ≤ 4 − 2√3 or k ≥ 4 + 2√3.

3️⃣ Common Types of Parameter-Based Questions
  • Find range of parameter for which roots are real/equal/imaginary.
  • Find parameter when roots satisfy a condition (e.g., one root is double the other).
  • Find parameter so that both roots are positive, negative, or lie in a given interval.
  • Find parameter when sum or product of roots equals a given value.
4️⃣ Key Strategy
  1. Write the quadratic equation clearly and identify a, b, c.
  2. Compute discriminant D = b² − 4ac.
  3. Apply the required condition (real, equal, positive roots, etc.).
  4. Simplify to get the range or value of the parameter.
Quick Tip for JEE & Advanced Exams: - Always check whether parameter appears in a, b, or c — this decides how the discriminant behaves.
- Combine sign of a and nature of roots for inequality-based parameter problems.
- Practice previous year JEE questions — these appear frequently in Parametric Condition and Root Nature chapters.

Transformation of Equations

The transformation of equations helps us form a new quadratic equation when the roots of the original equation are changed or modified in a specific way — such as taking reciprocals, squares, multiples, or shifts of the roots.

1️⃣ Standard Equation

Let the original quadratic equation be:

ax² + bx + c = 0   having roots   α and β
2️⃣ Transformation Rules (Common Cases)
New Roots Transformation Formula New Equation
Reciprocals (1/α, 1/β) Interchange coefficients of x² and constant term c·x² + b·x + a = 0
Squares (α², β²) Use sum = (α + β)² − 2αβ, product = (αβ)² x² − [(α + β)² − 2αβ]x + (αβ)² = 0
Multiples (kα, kβ) Replace x with x/k a(x/k)² + b(x/k) + c = 0 → multiply by k²
Shifted Roots (α + h, β + h) Replace x with (x − h) a(x − h)² + b(x − h) + c = 0
Inverted Shift (1/(α + h), 1/(β + h)) Apply both reciprocal and shift transformations stepwise Complex but solvable via substitution method
3️⃣ Example Problem

The equation x² − 5x + 6 = 0 has roots α and β. Find the equation whose roots are 2α and 2β.

Solution:

  • Given: α + β = 5, αβ = 6
  • New roots → 2α, 2β
  • Sum of new roots = 2(α + β) = 10
  • Product of new roots = 4(αβ) = 24

✅ Required equation → x² − 10x + 24 = 0

4️⃣ General Method
  1. Find α + β and αβ from the given equation.
  2. Apply transformation to get sum’ and product’ of new roots.
  3. Form the new equation using:
    x² − (sum’)x + (product’) = 0
Quick Tip for JEE & Boards: - Transformations test your understanding of relations between roots and coefficients.
- Most problems can be solved in 2–3 steps if you recall the formula Sum = α + β, Product = αβ.
- Practice transformations like reciprocal, shifted, scaled, squared roots frequently — they are high-scoring.

Graphical Solutions

The graphical method provides a visual understanding of how quadratic equations relate to their solutions (roots). By analyzing the intersection points of curves, students can understand the nature and number of roots clearly.

1️⃣ Basic Concept

A quadratic equation ax² + bx + c = 0 can be represented as a parabola:

  • Draw the curve y = ax² + bx + c.
  • The x-intercepts (points where the parabola cuts the x-axis) give the roots of the equation.
  • The number of intersections = number of real roots.
2️⃣ Nature of Roots (Visual Meaning)
Condition Graph Behavior Nature of Roots
D = b² − 4ac > 0 Parabola cuts x-axis at two distinct points Two distinct real roots
D = 0 Parabola touches x-axis at one point (vertex on x-axis) Two equal real roots
D < 0 Parabola lies completely above or below x-axis No real roots (complex roots)
3️⃣ Example

Consider the equation x² − 4x + 3 = 0.

  • Graph of y = x² − 4x + 3 is a parabola opening upwards.
  • It cuts the x-axis at x = 1 and x = 3.
  • Hence, the roots are 1 and 3.

✅ Graph confirms two real and distinct roots.

4️⃣ Intersection with Other Curves

Sometimes quadratic equations are formed by the intersection of two curves, such as:

  • y = ax² + bx + c and y = k → gives real roots where they meet horizontally.
  • y₁ = x² + 2x and y₂ = 3x + 1 → intersection points give the roots of x² − x − 1 = 0.
Quick Tip for Students: - Always link the Discriminant (D) to the graph — it predicts intersection count.
- a > 0 → parabola opens upward; a < 0 → downward.
- In JEE & Boards, graphical interpretation questions test conceptual clarity more than calculation.

Applications in Real Life & Geometry

Quadratic equations play a vital role in understanding real-world phenomena — from motion and optimization to geometric design. They appear wherever quantities vary **non-linearly**.

1️⃣ Physics & Engineering

Many motion and energy problems naturally form quadratic relationships.

  • Projectile Motion: The path of a projectile is a parabola defined by y = ax² + bx + c. Example: Finding maximum height or time of flight.
  • Free Fall & Acceleration: Equations like s = ut + ½at² are quadratic in time (t).
  • Electrical Circuits: Power and resistance relations often lead to quadratic forms, e.g., solving for current in P = I²R.
2️⃣ Economics & Business

Quadratic models are used to study profit, cost, and revenue functions — especially for optimization.

  • Profit Maximization: Profit function P(x) = -ax² + bx + c has a maximum at x = -b / 2a.
  • Cost Optimization: Used to minimize production cost and determine break-even points.
  • Demand Curves: Non-linear demand and supply relationships are modeled using quadratic equations.
3️⃣ Geometry & Coordinate Applications
  • Finding Tangents and Normals: Tangent equations to a parabola y² = 4ax or y = ax² involve quadratic concepts.
  • Area and Length Problems: Many area/distance optimization problems lead to quadratic equations (e.g., minimum distance from a point to a line).
  • Circle & Parabola Intersections: Their intersection points are obtained by solving quadratic equations.
4️⃣ Computer Graphics & Technology

Quadratic equations are used in digital modeling, animation, and trajectory calculations.

  • Bezier curves and parabolic shapes in 3D graphics.
  • Camera focusing and depth simulation use quadratic interpolation.
  • Machine learning algorithms and regression analysis often involve quadratic cost functions.
Key Insight: Quadratic equations are not just algebraic exercises — they’re the foundation of optimization, motion, and geometric modeling across science and engineering.

Practice Problems

Strengthen your understanding by solving carefully selected problems covering every major concept in quadratic equations. Includes MCQs, Integer-type, and Subjective questions with step-by-step solutions.

📘 Concept Areas Covered
  • Standard Form & Basic Concepts
  • Nature of Roots & Discriminant
  • Relations Between Roots & Coefficients
  • Formation & Transformation of Equations
  • Maximum–Minimum Value Problems
  • Graph and Sign Analysis
  • Parametric & Condition-Based Equations

Solution:

For equal roots, Discriminant D = 0

D = k² - 4(1)(16) = 0 ⇒ k² = 64 ⇒ k = ±8

✅ k = 8 or k = -8

Solution:

Let roots be α and 2α.

Sum = α + 2α = 3α = -b/a = -7/3 ⇒ α = -7/9

Product = α × 2α = 2α² = k/a = k/3

⇒ 2(49/81) = k/3 ⇒ k = 98/27

✅ k = 98/27

Solution:

For real and distinct roots, D > 0

D = k² – 4(k – 5) = k² – 4k + 20 > 0

Since (k – 2)² + 16 > 0 for all k, D is always positive.

So, all real k satisfy the condition ⇒ No restriction.

But since it asks for integer k, any integer is valid.

✅ All integers (no limit)

Solution:

Given α + β = 7, αβ = 10

New roots: α + 3, β + 3

Sum = (α + 3) + (β + 3) = 7 + 6 = 13

Product = (α + 3)(β + 3) = αβ + 3(α + β) + 9 = 10 + 21 + 9 = 40

Equation: x² – 13x + 40 = 0

✅ Required equation: x² – 13x + 40 = 0

Pro Tip: Practice at least 20 problems daily — mixing all levels (easy to advanced). Try solving without looking at formulas first to build real problem-solving intuition.

Summary of Quadratic Equations

Here’s a complete quick revision of all important concepts, formulas, and results from the chapter Quadratic Equations — useful for last-minute preparation before JEE, NEET, and board exams.

📘 1️⃣ Standard Form

The general form of a quadratic equation is: ax² + bx + c = 0, where a ≠ 0.

It represents a parabola in the Cartesian plane.

📗 2️⃣ Nature of Roots

Given discriminant D = b² − 4ac:

  • D > 0 → Two distinct real roots
  • D = 0 → Two equal real roots
  • D < 0 → Two complex conjugate roots
📙 3️⃣ Formula for Roots

The roots of ax² + bx + c = 0 are given by:

x = (−b ± √(b² − 4ac)) / 2a

📒 4️⃣ Relations Between Roots and Coefficients
  • Sum of roots (α + β) = −b / a
  • Product of roots (αβ) = c / a

These are known as Vieta’s Relations.

📕 5️⃣ Vertex and Extreme Values
  • Vertex of parabola: (−b / 2a, (4ac − b²) / 4a)
  • If a > 0 → Vertex is minimum point
  • If a < 0 → Vertex is maximum point
📓 6️⃣ Graphical Representation

The graph of y = ax² + bx + c is a parabola:

  • Opens upwards if a > 0
  • Opens downwards if a < 0
  • Axis of symmetry: x = −b / 2a
📔 7️⃣ Common Applications
  • Projectile motion, area optimization, and cost-profit analysis
  • Finding tangent points on curves in coordinate geometry
  • Used in real-life modeling, statistics, and physics
💡 Quick Recap Tips:
  • Always check discriminant before solving.
  • Memorize α + β = −b/a and αβ = c/a — they simplify many problems.
  • Convert real-life problems into quadratic form to analyze mathematically.
  • In exams, use vertex formula for quick max/min questions.
✅ You’re Now Ready for Any Quadratic Equation Problem!

Revise once before your test, and then jump into solving PYQs and mock tests.