IGCSE Mathematics: Algebra — Linear and Quadratic Equations, Inequalities, Sequences

TL;DR

Algebra is all about finding unknown values using equations and rules. Linear and quadratic equations help solve for single variables, while inequalities show a range of possible values. Sequences involve patterns that let you predict future terms.

1. The Mental Model

Think of algebra as detective work: you're given clues (equations, inequalities, sequence patterns) and you need to find the missing information (the unknown values or the rule). It's a systematic way to solve puzzles.

2. The Core Material

Linear Equations

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Linear equations are the simplest type, where the highest power of the unknown variable (usually 'x') is 1. Your goal is to isolate 'x'.

How to Solve:
1. Expand brackets: If there are any.
2. Collect like terms: Get all 'x' terms on one side and all constant terms on the other.
3. Perform inverse operations: If 'x' is multiplied by a number, divide. If a number is added to 'x', subtract.

Example:
Solve $3(x - 2) = x + 4$
$3x - 6 = x + 4$ (Expand brackets)
$3x - x = 4 + 6$ (Collect like terms)
$2x = 10$
$x = 5$

Quadratic Equations

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Quadratic equations have the highest power of 'x' as 2 (e.g., $ax^2 + bx + c = 0$). They usually have two solutions.

Methods to Solve:
1. Factorising: If you can express the quadratic as a product of two linear factors.
* Example: $x^2 + 5x + 6 = 0 \implies (x+2)(x+3) = 0 \implies x = -2$ or $x = -3$.
2. Quadratic Formula: Always works for any quadratic equation in the form $ax^2 + bx + c = 0$.
* $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
3. Completing the Square: Useful for finding the turning point of a parabola or deriving the quadratic formula.

Inequalities

Inequalities use symbols like $<, >, \le, \ge$ to show a range of values. You solve them much like linear equations, but with one crucial difference:

Key Rule: If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Example:
Solve $2x - 3 < 7$
$2x < 7 + 3$
$2x < 10$
$x < 5$

Solve $-3x + 1 \ge 10$
$-3x \ge 10 - 1$
$-3x \ge 9$
$x \le -3$ (Divided by -3, so reversed the sign)

Sequences

A sequence is a list of numbers that follow a specific pattern or rule.

Types of Sequences:
1. Arithmetic Sequences: Have a common difference between consecutive terms.
* General term: $T_n = a + (n-1)d$
* $T_n$ = the $n$-th term
* $a$ = the first term
* $d$ = the common difference
* $n$ = term number
2. Geometric Sequences: Have a common ratio between consecutive terms. (Less common in basic IGCSE, but good to know.)
* General term: $T_n = ar^{n-1}$
* $r$ = common ratio
3. Quadratic Sequences: The differences between consecutive terms aren't constant, but the differences of the differences are.
* The general term will be in the form $An^2 + Bn + C$. You find A, B, and C by setting up simultaneous equations.

Finding the $n$-th Term (General Rule) for Arithmetic Sequences:

graph TD
    A["Start: Sequence (e.g., 2, 5, 8, 11...)"] --> B["Find Common Difference (d)"]
    B --> C{"Is it Arithmetic?"}
    C -- "Yes" --> D["Write T_n = dn + c"]
    C -- "No" --> E["Consider Quadratic/Other"]
    D --> F["Substitute a known term to find c"]
    F --> G["Final Rule T_n = dn + c"]

Simultaneous Equations

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These involve two or more equations with two or more unknown variables (e.g., 'x' and 'y'). You need to find values that satisfy all equations.

Methods to Solve:
1. Substitution: Solve one equation for one variable, then substitute that expression into the other equation.
2. Elimination: Multiply equations by constants to make the coefficients of one variable the same (or additive inverses), then add or subtract the equations to eliminate that variable.

Example (Elimination):
1. $2x + y = 7$
2. $x - y = 2$

Add (1) and (2):
$(2x + y) + (x - y) = 7 + 2$
$3x = 9$
$x = 3$

Substitute $x=3$ into (2):
$3 - y = 2$
$y = 1$
Solution: $x=3, y=1$

3. Worked Example

Let's solve a quadratic equation using the formula.

Problem: Solve $2x^2 - 5x - 3 = 0$

Solution:
1. Identify $a, b, c$: In $2x^2 - 5x - 3 = 0$, we have $a=2$, $b=-5$, $c=-3$.
2. Write down the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
3. Substitute the values:
$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}$
$x = \frac{5 \pm \sqrt{25 - (-24)}}{4}$
$x = \frac{5 \pm \sqrt{25 + 24}}{4}$
$x = \frac{5 \pm \sqrt{49}}{4}$
$x = \frac{5 \pm 7}{4}$
4. Calculate the two possible solutions:
$x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3$
$x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$

So, the solutions are $x=3$ and $x=-\frac{1}{2}$.

4. Key Takeaways

• Always aim to isolate the unknown variable when solving equations and inequalities.
• Remember to reverse the inequality sign when multiplying or dividing by a negative number.
• Quadratic equations usually have two solutions, which can be found by factorising or using the quadratic formula.
• For arithmetic sequences, the common difference ($d$) is key to finding the $n$-th term.
• Simultaneous equations require finding a single pair of values that satisfy both equations.

Common Mistakes to Avoid:
- Forgetting to distribute a negative sign when expanding brackets like $-(x-3)$.
- Not reversing the inequality sign when multiplying/dividing by a negative number.
- Making arithmetic errors when substituting values into the quadratic formula.
- Confusing the common difference ($d$) with the common ratio ($r$) in sequences.

5. Now Try It

Spend 15 minutes trying to solve the following problems:
1. Solve for $x$: $5(x+1) - 2x = 2(x+7)$
2. Find the values of $x$ for which: $3x^2 + 10x - 8 = 0$ (Try factorising if you can, otherwise use the formula!)
3. Find the $n$-th term of the sequence: $7, 11, 15, 19, \dots$
4. Solve the simultaneous equations: $3x - 2y = 11$ and $x + 2y = 9$

You'll know you've succeeded if you can get correct answers for all four problems, showing your working clearly for each step.