Quadratic Equations | Class 10 Mathematics Notes

4.2 Quadratic Equations

This section formally defines a quadratic equation as an equation in the variable x of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. Any polynomial equation of degree 2 can be expressed in this standard form by arranging terms in descending order of degree. Examples such as 2x² + x - 300 = 0, 2x² - 3x + 1 = 0, 4x - 3x² + 2 = 0, and 1 - x² + 300 = 0 are shown to be quadratic equations after rewriting in standard form.

The section also illustrates how quadratic equations arise in various real-life contexts through two detailed examples. In the first example, John and Jivanti together have 45 marbles. After losing 5 each, the product of their remaining marbles is 124. By letting John's initial marbles be x, Jivanti's are 45 - x, and their remaining marbles are (x - 5) and (40 - x) respectively. The product equation leads to a quadratic equation x² - 45x + 324 = 0.

In the second example, a cottage industry produces x toys daily, with the cost per toy being (55 - x) rupees. The total cost is 750 rupees, leading to the quadratic equation x² - 55x + 750 = 0.

Another example checks whether given equations are quadratic by simplifying and rewriting them in standard form. This highlights the importance of simplification before classification. The section ends with an exercise to practice identifying quadratic equations and forming them from word problems.

📊 Diagram: Fig. 4.1

🧪 Activity: Example 1: Representing real-life situations as quadratic equations.

🔗 Connection: This section sets the foundation for solving quadratic equations, which is addressed in the next section.