Class 10th Quadratic Equations One Shot Class 10 Maths Chapter 4 Shobhit Nirwan

Quadratic Equations: Solving Techniques and Applications

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. The word "quadratic" comes from the Latin word "quadratus" meaning square, because the variable is squared (raised to the second power). Quadratic equations appear throughout physics, engineering, economics, and everyday problem-solving — from calculating the trajectory of a ball thrown in the air to determining the maximum profit in a business model. In the CBSE Class 10 curriculum, Chapter 4 builds directly on the linear equations studied in earlier chapters and lays the foundation for understanding higher-degree polynomials, the discriminant, and the nature of roots.

The most reliable method for solving any quadratic equation is the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. This formula is derived by completing the square on the general form ax² + bx + c = 0, and it works for every quadratic equation, whether the roots are rational, irrational, or complex. The expression under the square root, D = b² − 4ac, is called the discriminant, and it determines the nature of the roots without actually solving the equation. If D > 0, the equation has two distinct real roots. If D = 0, the equation has two equal real roots (also called repeated roots), meaning the parabola touches the x-axis at exactly one point. If D < 0, the equation has no real roots — the parabola does not intersect the x-axis at all. Before applying the formula, students should check whether the equation can be solved by factorisation, which is faster when the coefficients are simple integers. The method of completing the square is also important conceptually because it reveals how the quadratic formula is derived and helps in understanding the vertex form of a parabola.

A quadratic equation ax² + bx + c = 0 can also be represented graphically as a parabola on the coordinate plane. The coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0). The vertex of the parabola, which represents the maximum or minimum value of the quadratic function, is located at x = −b/(2a). The roots of the equation correspond to the x-intercepts of the parabola — the points where it crosses the x-axis. The relationship between the roots and coefficients is given by Vieta's formulas: the sum of the roots equals −b/a and the product of the roots equals c/a. Word problems involving quadratic equations frequently appear in board examinations — common types include problems on areas (where length and breadth are expressed as linear expressions in x), motion under gravity (where height is a quadratic function of time), and number puzzles (where consecutive integers or digits satisfy a quadratic condition).

• A quadratic equation in standard form is ax² + bx + c = 0 with a ≠ 0; it always has at most two real roots.
• The discriminant D = b² − 4ac determines the nature of roots: D > 0 gives two distinct real roots, D = 0 gives equal roots, D < 0 gives no real roots.
• The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves every quadratic equation.
• Factorisation is the quickest method when roots are rational; completing the square reveals the derivation of the formula.
• Sum of roots = −b/a and product of roots = c/a (Vieta's formulas); the vertex of the parabola is at x = −b/(2a).