Hey guys! Ready to dive into the world of algebra? If you're a Form 4 student, you're in the right place! Algebra can seem a bit intimidating at first, but trust me, with practice, it becomes a lot easier. This guide is packed with examples and practice questions to help you ace your algebra tests. We'll go through various topics, breaking down complex concepts into manageable chunks. So, grab your notebooks and let's get started!

Solving Linear Equations

Alright, let's kick things off with solving linear equations. This is like the bread and butter of algebra – you'll use this skill everywhere! A linear equation is an equation that looks like this: ax + b = c, where a, b, and c are constants, and x is the variable we want to find. The goal is always to isolate x on one side of the equation. Sounds simple, right? It totally is!

Let's start with a basic example: 2x + 3 = 7. Here's how we'd solve it:

1. Subtract 3 from both sides: This gets rid of the '+ 3' on the left side. So, we have 2x = 4.
2. Divide both sides by 2: This isolates x. So, x = 2.

See? Easy peasy! Now, let's amp up the difficulty a bit with a slightly more challenging problem. What about 3(x - 2) + 5 = 14?

1. Expand the brackets: Multiply the 3 by everything inside the parentheses. This gives us 3x - 6 + 5 = 14.
2. Simplify: Combine the constants. So, we get 3x - 1 = 14.
3. Add 1 to both sides: This gives us 3x = 15.
4. Divide both sides by 3: Finally, we get x = 5.

Practice makes perfect, so here are a few practice questions for you to try:

• 4x - 1 = 11
• 5(x + 2) = 25
• 2x + 7 = 3x - 1

Remember to take your time and double-check your work. Solving linear equations is a fundamental skill, and mastering it will set you up for success in more advanced algebra topics. Don't worry if you struggle a bit at first; just keep practicing, and you'll get the hang of it.

Quadratic Equations: An Introduction

Next up, let's explore quadratic equations. These are equations that involve x raised to the power of 2 (i.e., x²). They're a bit more complex than linear equations, but still totally manageable. A quadratic equation generally takes the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Solving quadratic equations involves finding the values of x that satisfy the equation. There are several methods to solve these equations: factoring, completing the square, and using the quadratic formula. Let's delve into them!

Factoring

Factoring is a method where you try to express the quadratic equation as a product of two linear factors. For instance, consider the equation x² + 5x + 6 = 0. To factor this, we need to find two numbers that add up to 5 (the coefficient of x) and multiply to 6 (the constant term). The numbers 2 and 3 fit the bill because 2 + 3 = 5 and 2 × 3 = 6. Thus, we can factor the equation as (x + 2)(x + 3) = 0. Then, we set each factor equal to zero and solve for x: x + 2 = 0 gives us x = -2, and x + 3 = 0 gives us x = -3. So, the solutions to the equation are x = -2 and x = -3. Factoring is often the easiest method if the equation can be easily factored.

Completing the Square

Completing the square is a more general method that can be used for any quadratic equation. The basic idea is to manipulate the equation to create a perfect square trinomial (a trinomial that can be factored into the square of a binomial). Here's how it works:

1. Move the constant term to the right side of the equation. For example, if we have x² + 6x + 5 = 0, we move the 5 to get x² + 6x = -5.
2. Take half of the coefficient of x, square it, and add it to both sides of the equation. In our example, half of 6 is 3, and 3 squared is 9. So, we add 9 to both sides: x² + 6x + 9 = -5 + 9.
3. Factor the left side as a perfect square trinomial and simplify the right side: (x + 3)² = 4.
4. Take the square root of both sides: x + 3 = ±2.
5. Solve for x: x = -3 ± 2. This gives us two solutions: x = -1 and x = -5.

Quadratic Formula

The quadratic formula is a universal method that can solve any quadratic equation. The formula is: x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients from the standard form ax² + bx + c = 0. Simply plug in the values of a, b, and c into the formula and solve for x. For instance, if you have x² + 4x - 5 = 0, then a = 1, b = 4, and c = -5. Plugging these values into the formula gives us:

x = (-4 ± √(4² - 4 × 1 × -5)) / (2 × 1) x = (-4 ± √(16 + 20)) / 2 x = (-4 ± √36) / 2 x = (-4 ± 6) / 2

This yields two solutions: x = 1 and x = -5. The quadratic formula is super handy, especially when factoring or completing the square is tricky or impossible.

Inequalities in Algebra

Alright, let's switch gears and talk about inequalities. Inequalities are like equations, but instead of an equals sign (=), we use symbols like