Notes from Toppers
Detailed Notes on Quadratic Equations
1. Nature of Roots:
 Discriminant ($D = b^2  4ac$):
 The discriminant determines the number and nature of roots of a quadratic equation.
 If (D > 0), the equation has two distinct real roots.
 If (D = 0), the equation has one repeated real root (also called a double root).
 If (D < 0), the equation has no real roots (complex conjugate roots).
[Reference: NCERT Class 11, Chapter 4, “Quadratic Equations”]
 Conditions for Real and Distinct Roots, Equal Roots, and No Real Roots:
 For real and distinct roots, (D > 0).
 For equal roots, (D = 0).
 For no real roots, (D < 0).
[Reference: NCERT Class 11, Chapter 4, “Quadratic Equations”]
2. Relationship between Roots and Coefficients:

Vieta’s Formulas:
 If (\alpha) and (\beta) are the roots of the quadratic equation (ax^2 + bx + c = 0), then:
 Sum of roots: (\alpha + \beta = b/a)
 Product of roots: (\alpha \beta = c/a)
 If (\alpha) and (\beta) are the roots of the quadratic equation (ax^2 + bx + c = 0), then:

Connection between the Roots and the Coefficients ((a, b, c)) of the Quadratic Equation:
 The sum of the roots (\alpha + \beta) is equal to the negative of the coefficient of (x), i.e., (b/a).
 The product of the roots (\alpha \beta) is equal to the constant term (c/a).
[Reference: NCERT Class 11, Chapter 4, “Quadratic Equations”]
3. Quadratic Equations Reducible to Linear Equations:

Concept of Reducible Quadratic Equations (Factorizable into Linear Factors):
 A quadratic equation is said to be reducible if it can be expressed as the product of two linear factors.
 A quadratic equation is reducible if and only if its discriminant (D) is a perfect square.

Solving These Equations by Factorization:
 If a quadratic equation is reducible, it can be solved by factoring the quadratic expression into linear factors and setting each factor equal to zero.
[Reference: NCERT Class 11, Chapter 4, “Quadratic Equations”]
4. Solutions of Quadratic Equations:

Finding Roots Using the Quadratic Formula:
 The quadratic formula is:
$$x = \frac{b \pm \sqrt{b^2  4ac}}{2a}$$
 This formula provides the two solutions or roots of a quadratic equation.
 The quadratic formula is:
$$x = \frac{b \pm \sqrt{b^2  4ac}}{2a}$$

Square Root Method (for Equations of the Form (x^2 = a)):
 For equations of the form (x^2 = a), the solutions can be found by taking the square root on both sides:
 If (a ≥ 0), the solutions are (x = ±\sqrt{a}).
 If (a < 0), there are no real solutions since the square root of a negative number is not real.
 For equations of the form (x^2 = a), the solutions can be found by taking the square root on both sides:
[Reference: NCERT Class 11, Chapter 4, “Quadratic Equations”]
5. Word Problems:

Application of Quadratic Equations to Solve Word Problems Involving:
 Areas
 Volumes
 Distances
 Other reallife scenarios

Read the problem carefully to identify the quadratic relationship and set up the appropriate quadratic equation.

Solve the equation using the methods discussed earlier.

Interpret the solutions in the context of the problem.
[Reference: NCERT Class 11, Chapter 4, “Quadratic Equations”, and relevant chapters in Class 12]
6. Graphical Representation:
 Sketching the Graphs of Quadratic Equations:
 Plot the intercepts (points where the parabola intersects the (x) and (y) axes).
 Find the vertex (the point where the parabola changes direction).
 Use the axis of symmetry (a vertical line through the vertex) to help sketch the parabola.
[Reference: NCERT Class 11, Chapter 6, “Pair of Linear Equations in Two Variables”]
 Identifying the Vertex, Axis of Symmetry, Intercepts, and Other Key Features of the Parabola:
 The vertex is the point with minimum or maximum (y)coordinate on the parabola.
 The axis of symmetry is a vertical line passing through the vertex.
 The (x)intercepts are the points where the parabola intersects the (x)axis.
 The (y)intercept is the point where the parabola intersects the (y)axis.
[Reference: NCERT Class 11, Chapter 6, “Pair of Linear Equations in Two Variables”]
7. Applications in Calculus:
 Tangents and Normals to Parabolas:
 Tangents and normals are straight lines that touch a parabola at a specific point.
 The slope of a tangent to a parabola is given by the derivative of the quadratic function at that point.
 The slope of a normal to a parabola at a point is the negative reciprocal of the slope of the tangent at that point.
[Reference: NCERT Class 12, Chapter 6, “Application of Derivatives”]
 Maxima and Minima Problems Using the First and Second Derivatives of Quadratic Equations:
 The first derivative of a quadratic function is a linear function, and its second derivative is a constant.
 Use the first derivative to find the critical points (where the derivative is zero or undefined).
 Use the second derivative to determine whether the critical point is a maximum or a minimum.
[Reference: NCERT Class 12, Chapter 6, “Application of Derivatives”]
8. Inequalities Involving Quadratic Equations:
 Solving Quadratic Inequalities Graphically and Algebraically:
 Graph the quadratic function to visualize the regions where it is positive and negative.
 Use algebraic techniques such as factoring, test points, and sign analysis to determine the solution set of the inequality.
[Reference: NCERT Class 11, Chapter 6, “Inequalities”]
 Applications of Quadratic Inequalities in Optimization Problems:
 Quadratic inequalities can be used to model reallife situations involving constrained optimization problems.
 Examples include finding minimum or maximum values of functions, subject to certain conditions.
[Reference: NCERT Class 12, Chapter 6, “Application of Derivatives”]
9. Complex Numbers and Quadratic Equations:

Understanding the Concept of Complex Roots:
 Complex roots occur when the discriminant (D) of a quadratic equation is negative ((D < 0)).
 Complex roots are in conjugate pairs, meaning they have the same real part but differ in the imaginary part.

Solving Quadratic Equations with Complex Roots:
 Use the quadratic formula to find the complex conjugate roots.
 Express the complex roots in the form (\alpha ± \beta i), where (\alpha) and (\beta) are real numbers and (i = \sqrt{1}) is the imaginary unit.
[Reference: NCERT Class 11, Chapter 5, “Complex Numbers and Quadratic Equations”]
10. Miscellaneous Topics:
 Applications of Quadratic Equations in Conic Sections (Parabolas):
 Parabolas are conic sections that can be represented by quadratic equations.
 Study the various properties of parabolas, such as their vertex, focus, and directrix.
[Reference: NCERT Class 11, Chapter 11, “Conic Sections”]
 Distance and Midpoint Formula Involving Quadratic Expressions:
 Use quadratic expressions to find the distance between two points or the midpoint of a line segment when coordinates involve quadratic terms.
[Reference: NCERT Class 11, Chapter 7, “Straight Lines”]
Remember to practice solving a variety of quadratic equation problems, including theoretical questions, word problems, graphical analysis, and applicationbased scenarios, to strengthen your understanding of this topic for the JEE exam.