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Cambridge IGCSE Maths (0580) ยท Algebra & Graphs
Mini-Lesson

Algebra & Graphs

This mini-lesson covers Algebra & Graphs for Cambridge IGCSE Mathematics (0580): algebraic manipulation, equations, simultaneous & quadratic equations, inequalities, sequences, functions and graphs of functions (Extended content included).

Honest heads-up: these games test recall of methods and results, not full algebraic working. In the exam, show every line.

Answer as you go and collect โญ stars. Press Start.

Manipulation

Expanding & factorising

Expand brackets carefully, then collect like terms. Factorise by taking out the highest common factor, or expand a double bracket with a grid / FOIL.

(x + 3)(x + 5) = xยฒ + 8x + 15middle term = 3 + 5, last term = 3 ร— 5

Sign care: (x โˆ’ 4)(x + 4) = xยฒ โˆ’ 16, a difference of two squares โ€” the middle terms cancel.

Calculate

Your turn

1Expand and simplify (x + 6)(x + 2). It gives xยฒ + bx + 12. Type the value of b.
Hint: b = 6 + 2.
Linear equations

Solving linear equations

Do the same to both sides to keep the balance. Undo operations in reverse order (inverse of + is โˆ’, of ร— is รท).

Worked example

Solve 3x + 4 = 19.

3x = 15 โ†’ x = 5

Calculate

Your turn

2Solve 5x โˆ’ 7 = 28. Type the value of x.
Hint: add 7 (โ†’ 5x = 35), then divide by 5.
Simultaneous equations

Simultaneous equations

Two equations, two unknowns. Eliminate a variable by adding or subtracting (make coefficients match), or substitute.

Worked example

x + y = 10 and x โˆ’ y = 4. Add: 2x = 14 โ†’ x = 7. Then y = 3.

Solution: x = 7, y = 3 (check: 7+3 = 10 โœ“)

Calculate

Your turn

3Solve 2x + y = 13 and x + y = 8 simultaneously. Subtract the equations to find x.
Hint: (2x + y) โˆ’ (x + y) = 13 โˆ’ 8 gives x = 5.
Quadratics

Quadratic equations

Solve xยฒ + bx + c = 0 by factorising (find two numbers that multiply to c and add to b), or use the quadratic formula.

x = [โˆ’b ยฑ โˆš(bยฒ โˆ’ 4ac)] รท 2afor axยฒ + bx + c = 0
Worked example

xยฒ โˆ’ 7x + 12 = 0 โ†’ (x โˆ’ 3)(x โˆ’ 4) = 0 โ†’ x = 3 or x = 4

Calculate

Your turn

4Solve xยฒ โˆ’ 9x + 20 = 0. The solutions are x = 4 and x = k. Type the larger root k.
Hint: factors of 20 that add to 9 are 4 and 5, so (xโˆ’4)(xโˆ’5)=0.
Quick check

Quick check

?Which is a correct factorisation of xยฒ โˆ’ 25?
Inequalities

Inequalities

Solve like an equation, but reverse the inequality sign whenever you multiply or divide by a negative number.

Worked example

Solve 2x + 1 < 11 โ†’ 2x < 10 โ†’ x < 5.

Watch the flip: โˆ’x < 3 becomes x > โˆ’3 (divide by โˆ’1, flip the sign).

Calculate

Your turn

5Solve 4x + 3 โ‰ค 23. The solution is x โ‰ค k. Type k.
Hint: 4x โ‰ค 20, so x โ‰ค 5.
Sequences

Sequences & the nth term

For a linear (arithmetic) sequence the nth term is dn + (a โˆ’ d), where d is the common difference and a is the first term.

Worked example

5, 8, 11, 14โ€ฆ d = 3, first term 5, so nth term = 3n + 2.

Check n = 1: 3(1) + 2 = 5 โœ“

Calculate

Your turn

6A sequence has nth term 4n โˆ’ 1. Work out the 10th term.
Hint: substitute n = 10 into 4n โˆ’ 1.
Match game

Match the nth term

Tap a sequence, then its correct nth-term rule.

Functions & graphs

Functions & graphs

A function f(x) is a rule; f(3) means substitute x = 3. The inverse fโปยน(x) undoes it. Recognise graph shapes: linear (straight), quadratic (parabola), cubic, reciprocal (y = 1/x) and exponential.

Worked example

f(x) = 2x + 5. Then f(4) = 2(4) + 5 = 13.

Calculate

Your turn

7f(x) = 3x โˆ’ 4. Work out f(6).
Hint: 3 ร— 6 โˆ’ 4.
Quick check

Quick check

?What shape is the graph of y = xยฒ โˆ’ 4x + 3?
Sort it

Linear or non-linear?

Sort each equation by its graph shape. Tap it, then its box.

๐Ÿ“ Straight line

ใ€ฐ Curve

Recap

Key points

Manipulation: expand double brackets; factorise; difference of two squares.

Equations: keep the balance; simultaneous by elimination/substitution.

Quadratics: factorise or use x = [โˆ’b ยฑ โˆš(bยฒโˆ’4ac)]/2a.

Inequalities: flip the sign when รท or ร— by a negative.

Sequences: linear nth term dn + (a โˆ’ d).

Functions & graphs: f(x) substitution; recognise linear/quadratic/cubic/reciprocal/exponential shapes.

You've covered the Algebra & Graphs essentials for Cambridge IGCSE Maths (0580). Press Finish to see your score.

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