Key Concepts

Habari Mwanafunzi! Welcome to the World of Quadratics!

Ever thrown a ball to your friend? Or watched a javelin thrower at the school sports day? The path that the ball or the javelin takes through the air is a beautiful curve. That curve, my friend, is described by a special kind of maths called Quadratic Expressions! Today, we are going to learn the basic language of quadratics. Think of it as learning the ABCs before you can read a whole story. Are we together? Haya, let's begin!

First Things First: What is a "Quadratic" Expression?

The word "Quadratic" sounds complicated, but its secret is simple. It comes from the Latin word 'quadratus', which means "square". In mathematics, whenever you see 'quadratic', your mind should immediately think of something being squared, or raised to the power of 2.

So, a quadratic expression is simply a mathematical phrase whose most powerful, most important, "mkuu" term has a variable raised to the power of 2. This term is the boss!

The Official Uniform: The General Form

Every concept in math has a standard way of being written, like an official school uniform. For quadratic expressions, the general form is:

ax² + bx + c

Let's break down this uniform piece by piece:

• ax² is the Quadratic Term. It's the most important part!
  • 'x' is our variable.
  • The '²' is what makes it quadratic.
  • 'a' is the coefficient of x². It can be any number, but it cannot be zero. Why? If 'a' was 0, the x² term would disappear, and our expression would lose its quadratic superpower!
• bx is the Linear Term.
  • 'b' is the coefficient of x.
• c is the Constant Term.
  • It's just a plain number, standing on its own, with no variable attached.

Think of it like making a cup of tea. The ax² (tea leaves) is essential; without it, you don't have tea. The bx (sugar) is optional; some people like it, some don't. The c (milk) is also optional. But you MUST have the tea leaves!

Let's Practice: Identifying a, b, and c

This is like a game. We look at an expression and identify the values of a, b, and c. Sawa?

Example 1: 2x² + 5x + 3

- The number with x² is 'a'. So, a = 2
- The number with x is 'b'. So, b = 5
- The number on its own is 'c'. So, c = 3

Example 2: x² - 7x + 10

- The number with x² is 'a'. It looks empty, but there's a hidden 1! So, a = 1
- The number with x is 'b'. Don't forget the sign! So, b = -7
- The number on its own is 'c'. So, c = 10

Example 3: A tricky one! 4x² - 9

- The number with x² is 'a'. So, a = 4
- Is there a term with just 'x'? No! This means 'b' is hiding. So, b = 0
- The number on its own is 'c'. So, c = -9

Example 4: Rearrange it first! 12 + 5x² - 3x

Step 1: Write it in the general form (ax² + bx + c).
         5x² - 3x + 12

Step 2: Now identify a, b, and c.
         a = 5
         b = -3
         c = 12

A Real-Life Kenyan Example

Let's imagine a farmer, Mama Boke, from Kisii. She wants to fence a rectangular vegetable garden (shamba) next to a long river, so she only needs to fence three sides. She has 100 metres of fencing wire. She wants to find the biggest possible area for her vegetables.

Let's draw a small picture of the shamba:

    RIVER RRRRRRRRRRRRRRRRRRRRRRRR
    +---------------------------+
    |                           |
    |                           | x
    |      VEGETABLE GARDEN     |
    |                           |
    +---------------------------+
               Length

Let the two sides perpendicular to the river be 'x' metres each. The total wire is 100m. So the length of the side parallel to the river will be what's left over: 100 - x - x = 100 - 2x.

Now, we know the area of a rectangle is Length × Width.

Area = (100 - 2x) * x
Area = 100x - 2x²

Look at that! We have a quadratic expression: -2x² + 100x. Here, a = -2, b = 100, and since there is no constant term, c = 0. By studying this expression, Mama Boke can figure out the exact width 'x' that will give her the maximum possible area for her sukuma wiki! That's the power of quadratics!

Image Suggestion: A vibrant, sunlit digital painting of a Kenyan female farmer (Mama Boke) smiling proudly next to her lush vegetable garden (shamba) bordering a serene river. The style should be colourful and optimistic, showing rows of sukuma wiki and other greens. The fencing should be clearly visible on three sides of the rectangular plot.

Why is this Important?

Understanding these key concepts is the foundation for everything else we will do in this topic. Soon, we will learn how to:

• Factorise these expressions.
• Solve quadratic equations to find the value of 'x'.
• Draw graphs of these expressions, which create a beautiful 'U' shape called a parabola.

You have taken your first big step into a very powerful part of mathematics. Keep practicing identifying 'a', 'b', and 'c', and you'll be a master in no time. Kazi nzuri!