4. QUADRATIC EQUATIONS

In Chapter 2, you have studied different types of polynomials. One type was the quadratic polynomial of the form ax² + bx + c, where a ≠ 0. When we equate this polynomial to zero, we get a quadratic equation. Quadratic equations come up when we deal with many real-life situations. For instance, suppose a charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall? Suppose the breadth of the hall is x metres. Then, its length should be (2x + 1) metres. We can depict this information pictorially. Now, area of the hall = (2x + 1) * x = (2x² + x). So, 2x² + x = 300 (Given), therefore, 2x² + x – 300 = 0.

Quadratic equations are fundamental mathematical concepts represented as ax² + bx + c = 0. They are formed from quadratic polynomials. In this context, 'a' cannot be zero, as this would make it a linear equation instead. Quadratic equations represent various real-world scenarios, such as calculating dimensions of objects or optimizing situations. The example given illustrates how we can represent a problem involving a prayer hall’s dimensions with a quadratic equation.

Imagine trying to determine how much material you need for a tent. If the tent’s floor area must be a specific size, like 300 square meters, and there’s a relationship between its length and breadth (like the length being more than twice the breadth), you can use a quadratic equation to find the dimensions. You can visualize the situation with drawings or diagrams to understand the relationship better.

A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers, a ≠ 0. For example, 2x² + x – 300 = 0 is a quadratic equation. In fact, any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation.

A quadratic equation is a special type of polynomial equation of degree 2. In the standard form, it consists of three components: 'a' (the coefficient of x², which must not be zero), 'b' (the coefficient of x), and 'c' (the constant term). This form allows us to identify the equation easily and solve it using various methods.

Think of a quadratic equation like a balance scale that determines the weight (value) of a combination of items: the 'a' represents how heavy the x² item is, 'b' is like a lighter item, and 'c' is a fixed weight. If you know the total weight is zero, you can figure out the individual weights (roots) of x, just like solving for unknowns in a real situation.

Example 1: Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We want to find out how many marbles they had to start with. Solution: Let the number of marbles John had be x. Then the number of marbles Jivanti had = 45 – x. The number of marbles left with John after losing 5 is x – 5, and Jivanti has 40 – x. Therefore, their product (x – 5)(40 – x) = 124 gives the quadratic equation x² - 45x + 324 = 0. (ii) A cottage industry produces toys, with the cost of each toy at 55 minus the number produced, with a total cost of 750. The equation formed is x² - 55x + 750 = 0.

In this section, we look at how certain real-life situations translate into quadratic equations. John and Jivanti's marble situation showcases how two unknowns can create a quadratic when their combined relationships are expressed algebraically. Similarly, the toy production scenario illustrates the practical applications of quadratic equations in business settings, where production costs influence decision-making.

Consider planning a community event where you must determine how many tickets to sell for a function (John and Jivanti) and how ticket price influences total revenue (toys produced). By setting up equations reflecting these relationships, you can derive conclusions (like profit or attendance) in an organized manner using quadratic equations.

Example 2: Check whether the following are quadratic equations: (i) (x – 2)² + 1 = 2x – 3. LHS = (x – 2)² + 1 simplifies to a quadratic form; therefore, it's a quadratic equation. (ii) x(x + 1) + 8 = (x + 2)(x – 2) simplifies and shows it's not a quadratic equation. (iii) Similarly checks indicate whether equations hold quadratic forms, with necessary simplifications.

This chunk focuses on how to identify quadratic equations from given mathematical statements. To convert into a standard form, we often need to simplify the given expressions through algebraic manipulation. Quadratic checks reveal if expressions lead to a parabolic relationship based on their standard format.

Imagine receiving invitations to multiple events where you only want to attend those that will give you quadratic returns in enjoyment. Identifying which events (equations) fit your expectations (quadratic criteria) helps you decide where to invest your time.

In this chapter, you have studied the following points: 1. A quadratic equation in the variable x is of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. 2. A real number α is a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0. 3. If we can factorize ax² + bx + c into a product of two linear factors, the roots of the quadratic equation can be found by equating each factor to zero.

The summary reflects on the core components learned throughout the chapter. Key concepts include understanding the structure of quadratic equations, roots as solutions to these equations, and methods of solving. This recap reinforces knowledge, helping solidify foundational principles and prepare for more advanced problem-solving.

Think of this chapter as building a toolbox. Each point represents a different tool – knowing their functions (quadratic equations, roots, factorization, etc.) that can help you tackle various challenges in math, just like specific tools help in fixing different things around the house.

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