Quadratic Equations

4.1 Introduction

In this section, the concept of quadratic equations is introduced by linking it to quadratic polynomials studied earlier. A quadratic polynomial is of the form ax² + bx + c, where a ≠ 0. When this polynomial is set equal to zero, it forms a quadratic equation. Quadratic equations are important because they arise naturally in many real-life problems. For example, a charity trust wants to build a prayer hall of carpet area 300 square meters, with length one meter more than twice the breadth. If the breadth is x meters, then length is (2x + 1) meters, so the area is x × (2x + 1) = 2x² + x. Setting this equal to 300 gives the quadratic equation 2x² + x - 300 = 0. This shows how quadratic equations model practical problems involving dimensions and areas. Historically, quadratic equations have been studied by ancient civilizations. Babylonians knew how to solve problems equivalent to quadratic equations, such as finding two positive numbers with a given sum and product. Greek mathematician Euclid used geometric methods related to quadratic equations. Ancient Indian mathematicians like Brahmagupta gave explicit formulas for solving quadratic equations of the form ax² + bx = c. Later, Sridharacharya derived the quadratic formula by completing the square, a method still used today. Arab mathematician Al-Khwarizmi also studied quadratic equations extensively. Thus, quadratic equations have a rich historical background and are fundamental in mathematics and applications.

• A quadratic polynomial is of the form ax² + bx + c, with a ≠ 0.
• Setting the quadratic polynomial equal to zero forms a quadratic equation.
• Quadratic equations model real-life problems involving areas and dimensions.
• Example: Prayer hall area problem leads to quadratic equation 2x² + x - 300 = 0.
• Babylonians, Greeks, Indians, and Arabs contributed to solving quadratic equations historically.
• Sridharacharya derived the quadratic formula by completing the square.
• 📌 Quadratic polynomial: A polynomial of degree 2, ax² + bx + c with a ≠ 0.
• 📌 Quadratic equation: An equation obtained by equating a quadratic polynomial to zero.

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.

• A quadratic equation is of the form ax² + bx + c = 0 with a ≠ 0.
• Any polynomial equation of degree 2 can be rewritten in standard form.
• Real-life problems can be modeled as quadratic equations.
• Example problems involve marbles and production cost leading to quadratic equations.
• Simplification is necessary to identify whether an equation is quadratic.
• Not all equations that look quadratic are quadratic after simplification.
• 📌 Standard form: Quadratic equation arranged with terms in descending powers of x.
• 📌 Polynomial degree: Highest power of the variable in the polynomial.