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Mathematics · Class 10 · Chapter 20: Theory of Quadratic Equations

8 min

Key points

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0
The discriminant Δ = b² - 4ac determines the nature of roots
If Δ > 0, roots are real and distinct; if Δ = 0, roots are real and equal; if Δ < 0, roots are complex
Sum of roots = -b/a and Product of roots = c/a
Quadratic formula: x = (-b ± √(b² - 4ac))/2a
A quadratic equation can be solved by factorization, completing the square, or using the quadratic formula
The graph of a quadratic function is a parabola
Maximum and minimum values occur at vertex x = -b/2a

Definitions

Quadratic Equation: An equation of the second degree in one variable is called a quadratic equation. The general form is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Discriminant: The expression b² - 4ac is called the discriminant of the quadratic equation ax² + bx + c = 0. It is denoted by Δ (Delta).
Roots of Quadratic Equation: The values of the variable which satisfy the quadratic equation are called the roots or solutions of the equation.
Nature of Roots: The character of roots (real, equal, distinct, or complex) is determined by the value of discriminant.

Worked example

Define discriminant of a quadratic equation. Find the discriminant of 2x² - 5x + 3 = 0 and determine the nature of its roots. (5 marks)

Discriminant of quadratic equation ax² + bx + c = 0 is Δ = b² - 4ac. Here a = 2, b = -5, c = 3. Therefore Δ = (-5)² - 4(2)(3) = 25 - 24 = 1. Since discriminant is positive, the roots are real and unequal.

Common mistakes

Students forget that a ≠ 0 in the definition of quadratic equation
Wrong calculation of discriminant - forgetting the negative sign in -4ac
Confusing sum and product of roots formulas
Not simplifying the quadratic formula completely
Forgetting to write ± in the quadratic formula

Quick recap

Quadratic equation has form ax² + bx + c = 0 where a ≠ 0. Discriminant Δ = b² - 4ac determines nature of roots: Δ > 0 gives real and unequal roots, Δ = 0 gives real and equal roots, Δ < 0 gives complex roots. Sum of roots = -b/a, product of roots = c/a. Always identify coefficients clearly and use exact textbook definitions.

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