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Class 10 · Maths · Chapter 4 · NCERT Class 10 Mathematics

Quadratic Equations Class 10 Notes

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Chapter mind map

The whole chapter at a glance: the big idea, then each branch and what sits under it.

Quadratic Equations

A polynomial equation of degree two in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

  1. Standard Form and Identification

    Equations must be simplified to ax² + bx + c = 0 to verify their degree; terms may cancel out during expansion.

    • Non-zero Coefficient Requirement — The coefficient 'a' must be non-zero, otherwise the equation becomes linear (bx + c = 0).
    • Simplification Process — Expand brackets and combine like terms to reveal the true degree, as cubic or squared terms may disappear.

  2. Roots and Solutions

    A root α is a real number that satisfies the equation aα² + bα + c = 0, equivalent to the zeroes of a polynomial.

    • Maximum Two Roots — A quadratic equation has at most two roots, representing where the function crosses the x-axis.
    • Factorisation Method — Splitting the middle term 'b' into two numbers that add to 'b' and multiply to 'ac'.
    • Zero Product Property — If (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0 to find individual roots.

  3. The Quadratic Formula

    A direct calculation method: x = (-b ± √D) / (2a), where D is the discriminant b² - 4ac.

    • Formula Application — Used when factorisation is difficult; valid only if the discriminant is non-negative.
    • Historical Development — Brahmagupta provided an explicit formula in the 7th century; Sridharacharya derived the modern version c. 1025 C.E.

  4. Nature of Roots

    The discriminant D = b² - 4ac determines the number and type of solutions without solving the equation.

    • Two Distinct Real Roots — Occurs when D > 0; the parabola crosses the x-axis at two distinct points.
    • Two Equal Real Roots — Occurs when D = 0; also known as coincident roots where the parabola touches the x-axis once.
    • No Real Roots — Occurs when D < 0; the square root of a negative number is not real and the parabola never touches the x-axis.

  5. Mathematical Modelling

    Translating real-world scenarios into algebraic equations by defining variables and relationships.

    • Modelling Steps — Identify the unknown 'x', express other quantities in terms of 'x', and form an equation based on conditions.
    • Feasibility and Constraints — Reject roots that lack physical meaning (e.g., negative distance) and use D to check if a situation is possible.

Chapter notes

An in-depth study of quadratic equations, exploring their standard form, methods of solving through factorisation and the quadratic formula, and the nature of roots based on the discriminant.

01 · Explore

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which arises naturally in various real-life mathematical modeling scenarios.

In previous studies of polynomials, we encountered the quadratic polynomial of the form ax² + bx + c, where a is not equal to zero. When we equate such a polynomial to zero, we form a quadratic equation. This mathematical structure is essential for solving problems involving areas, trajectories, and optimization.

Historically, the development of quadratic equations spans several civilizations. The Babylonians solved problems equivalent to x² - px + q = 0, while Greek mathematician Euclid used geometric methods. Significant progress was made by ancient Indian mathematicians; Brahmagupta provided an explicit formula for ax² + bx = c in the 7th century, and Sridharacharya derived the quadratic formula we use today around 1025 C.E.

The standard form of a quadratic equation in the variable x is ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The condition a ≠ 0 is critical because if a were zero, the equation would become linear (bx + c = 0), losing its quadratic nature.

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Why must the coefficient 'a' be non-zero in the standard form ax² + bx + c = 0?

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NCERT reference: chapter PDF pages 1, 2.

02 · Explore

Identifying Quadratic Equations

Not every equation that contains an x² term is a quadratic equation, and some equations that look cubic may actually be quadratic after simplification.

To determine if an equation is quadratic, it must be simplified into the form ax² + bx + c = 0. This often requires expanding brackets, combining like terms, and moving all terms to one side of the equality.

Consider the equation x(x + 1) + 8 = (x + 2)(x - 2). At first glance, it appears quadratic. However, expanding both sides gives x² + x + 8 = x² - 4. When we subtract x² from both sides, the squared terms cancel out, leaving x + 12 = 0. Since the degree is now 1, it is not a quadratic equation.

Conversely, an equation like (x + 2)³ = x³ - 4 might look cubic. Expanding the left side gives x³ + 6x² + 12x + 8 = x³ - 4. The x³ terms cancel, resulting in 6x² + 12x + 12 = 0, which is a quadratic equation.

Verifying a Quadratic Form

(x - 2)² + 1 = 2x - 3

Expand the LHS: x² - 4x + 4 + 1 = x² - 4x + 5. Equate to RHS: x² - 4x + 5 = 2x - 3. Move all terms to LHS: x² - 6x + 8 = 0. This matches ax² + bx + c = 0 where a=1, b=-6, c=8.

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Is x² + 3x + 1 = (x - 2)² a quadratic equation?

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NCERT reference: chapter PDF pages 3, 4.

03 · Explore

Mathematical Modelling with Quadratic Equations

Real-world situations can be translated into quadratic equations by defining a variable and expressing other quantities in terms of that variable.

When modeling a situation, we identify the unknown quantity to be found and represent it as 'x'. We then use the given conditions to form an algebraic expression. For example, if the area of a rectangular plot is 528 m² and the length is one more than twice its breadth, we let breadth be x. Then length is (2x + 1).

The area is calculated as Length × Breadth, so x(2x + 1) = 528. This simplifies to 2x² + x - 528 = 0. Solving this quadratic equation allows us to find the physical dimensions of the plot.

Another common scenario involves the product of consecutive integers. If two consecutive positive integers have a product of 306, we let the integers be x and (x + 1). The equation becomes x(x + 1) = 306, or x² + x - 306 = 0.

Steps to Represent Situations Mathematically

  1. 1

    Identify Variable

    Choose 'x' to represent the unknown quantity (e.g., breadth or number of items).

  2. 2

    Express Others

    Write other related quantities in terms of 'x' based on the problem's conditions.

  3. 3

    Form Equation

    Use a relationship like Area = L × B or Total Cost = Count × Rate to set up the equation.

  4. 4

    Standard Form

    Simplify the expression into ax² + bx + c = 0.

The process of converting a word problem into a solvable quadratic equation.

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If the sum of two numbers is 27 and their product is 182, what is the quadratic equation representing this?

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NCERT reference: chapter PDF pages 2, 3, 4, 5.

04 · Explore

Roots of Quadratic Equations

A root of a quadratic equation is a value that, when substituted for the variable, makes the equation true.

A real number α is called a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0. This is conceptually identical to the 'zeroes' of the quadratic polynomial we studied in Chapter 2. If α is a root, we say that x = α satisfies the equation.

Since a quadratic polynomial has a degree of 2, it can have at most two roots. These roots represent the points where the corresponding quadratic function would cross the x-axis.

Finding the roots is the primary goal of solving the equation. We can verify if a value is a root by direct substitution. For example, in 2x² - 3x + 1 = 0, substituting x = 1 gives 2(1)² - 3(1) + 1 = 2 - 3 + 1 = 0. Thus, 1 is a root.

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What is the relationship between the zeroes of a polynomial p(x) and the roots of the equation p(x) = 0?

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NCERT reference: chapter PDF page 5.

The rest of this chapter

Keep reading Quadratic Equations, free

  1. Locked: 1. Solution by Factorisation
  2. Locked: 2. The Quadratic Formula
  3. Locked: 3. Nature of Roots and the Discriminant
  4. Locked: 4. Applications and Problem Solving

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