Topic 2: Algebra and graphs

Algebra is the language that lets you describe patterns and relationships using letters instead of fixed numbers. Topic 2 of the Cambridge IGCSE Mathematics (0580) syllabus is the largest and most heavily examined area, and almost every later topic leans on the skills you build here. The work splits into two connected halves. The first is symbolic manipulation: simplifying, expanding, factorising, solving and rearranging. The second is graphical thinking: turning equations into curves, reading information from those curves, and describing how one quantity varies with another. This article walks through both halves in order, building from the most basic operations up to functions, graphs and variation. Work through every worked example with a pen, because algebra is a skill you learn by doing, not by reading.

Simplifying, expanding and factorising

Every algebra question begins with tidy manipulation. To simplify, you collect like terms, which are terms with exactly the same letters raised to the same powers. For example $3x + 5y - x + 2y = 2x + 7y$. When multiplying and dividing, use the index laws: $a^{m} \\times a^{n} = a^{m+n}$, $a^{m} \\div a^{n} = a^{m-n}$, and $(a^{m})^{n} = a^{mn}$. So $2x^{3} \\times 4x^{2} = 8x^{5}$. Expanding means removing brackets by multiplying out. A single bracket uses the distributive law: $3(2x - 5) = 6x - 15$. For two brackets, multiply every term in the first by every term in the second (often remembered as FOIL): $(x + 3)(x - 4) = x^{2} - 4x + 3x - 12 = x^{2} - x - 12$. A very useful special case is the perfect square $(x + a)^{2} = x^{2} + 2ax + a^{2}$ and the difference of two squares $(x + a)(x - a) = x^{2} - a^{2}$. Factorising is the reverse of expanding: you write an expression as a product of factors. First always look for a common factor: $6x^{2} + 9x = 3x(2x + 3)$. For a quadratic $x^{2} + bx + c$, find two numbers that multiply to $c$ and add to $b$. For $x^{2} - x - 12$ the numbers are $-4$ and $3$, giving $(x - 4)(x + 3)$. Recognise the difference of two squares too: $9x^{2} - 25 = (3x + 5)(3x - 5)$.

Solving linear equations and rearranging formulae

A linear equation has the unknown only to the first power, with no $x^{2}$ or higher term. To solve, do the same operation to both sides until the unknown stands alone. For $5x - 7 = 2x + 8$, subtract $2x$ from both sides to get $3x - 7 = 8$, add $7$ to get $3x = 15$, then divide by $3$ to get $x = 5$. Always substitute your answer back to check it works. When an equation contains fractions, multiply every term by the lowest common denominator to clear them. For $\\frac{x}{3} + \\frac{x}{4} = 7$, multiply through by $12$ to get $4x + 3x = 84$, so $7x = 84$ and $x = 12$. Rearranging a formula, also called changing the subject, uses exactly the same balancing rules but the target letter is buried among others. To make $r$ the subject of $A = \\pi r^{2}$, divide both sides by $\\pi$ to get $\\frac{A}{\\pi} = r^{2}$, then take the positive square root: $r = \\sqrt{\\frac{A}{\\pi}}$. When the wanted letter appears twice, gather those terms on one side and factorise it out. For $ay + 3 = by + c$, write $ay - by = c - 3$, factorise to $y(a - b) = c - 3$, then $y = \\frac{c - 3}{a - b}$.

Quadratic equations and the formula

A quadratic equation has the form $ax^{2} + bx + c = 0$ with $a$ not equal to $0$. There are three main solving methods. If the quadratic factorises neatly, use factorising and then the fact that a product is zero only when a factor is zero. For $x^{2} - x - 12 = 0$, factorise to $(x - 4)(x + 3) = 0$, giving $x = 4$ or $x = -3$. When factorising is awkward, use the quadratic formula, which solves any quadratic: $x = \\frac{-b \\pm \\sqrt{b^{2} - 4ac}}{2a}$. Read off $a$, $b$ and $c$ carefully, keeping their signs. For $2x^{2} + 3x - 4 = 0$ you get $x = \\frac{-3 \\pm \\sqrt{9 + 32}}{4} = \\frac{-3 \\pm \\sqrt{41}}{4}$, which gives $x = 0.85$ or $x = -2.35$ to two decimal places. The exam usually tells you to give answers to two decimal places when the roots are not whole numbers, which is a signal that the formula is expected. The quantity $b^{2} - 4ac$ under the square root is the discriminant. If it is positive there are two real solutions, if it is zero there is one repeated solution, and if it is negative there are no real solutions because you cannot take the square root of a negative number. The third method, completing the square, rewrites the quadratic as $(x + p)^{2} + q$ and is useful for finding the turning point of a parabola.

Simultaneous equations and inequalities

Simultaneous equations are two equations in two unknowns that are true at the same time. Two methods are common. Elimination adds or subtracts the equations to remove one unknown. For $3x + 2y = 16$ and $x - 2y = 0$, adding them removes $y$ to give $4x = 16$, so $x = 4$, and then $y = 2$. Substitution rearranges one equation and inserts it into the other; this is essential when one equation is non-linear, such as a line meeting a curve. For $y = x + 1$ and $y = x^{2} - 1$, substitute to get $x^{2} - 1 = x + 1$, so $x^{2} - x - 2 = 0$, giving $x = 2$ or $x = -1$ with matching $y$ values. Inequalities use the symbols $\\lt$, $\\gt$, $\\leq$ and $\\geq$, and you solve them almost exactly like equations. The one crucial rule is that multiplying or dividing both sides by a negative number reverses the inequality. So from $-2x \\gt 6$ you divide by $-2$ and flip to get $x \\lt -3$. Solutions are often shown on a number line, with an open circle for a strict inequality and a filled circle for $\\leq$ or $\\geq$. In graphical work you may shade regions defined by several inequalities to find the feasible area.

Sequences and the nth term

A sequence is an ordered list of numbers called terms, and the position of a term is its term number $n$. A linear or arithmetic sequence increases by a constant common difference $d$. Its nth term is $a + (n - 1)d$, where $a$ is the first term, but at IGCSE it is often quicker to write it as $dn + c$. For the sequence $5, 8, 11, 14, \\ldots$ the difference is $3$, so the rule starts $3n$, and since $3 \\times 1 = 3$ but the first term is $5$ you add $2$, giving the nth term $3n + 2$. A geometric sequence multiplies by a constant common ratio $r$ each time, so its nth term is $ar^{n-1}$. For $2, 6, 18, 54, \\ldots$ the ratio is $3$ and the nth term is $2 \\times 3^{n-1}$. You should also recognise quadratic sequences, where the second differences are constant. For $2, 6, 12, 20, \\ldots$ the first differences are $4, 6, 8$ and the second differences are all $2$. A constant second difference of $2k$ means the $n^{2}$ coefficient is $k$; here $k = 1$, so the nth term is $n^{2} + n$. Other patterns worth knowing on sight are the square numbers $n^{2}$, the cube numbers $n^{3}$, and the triangular numbers $\\frac{n(n + 1)}{2}$.

Functions: notation, composite and inverse

A function is a rule that takes an input and produces exactly one output. We write it as $f(x)$, read as f of x, where $x$ is the input. To evaluate, substitute the number. If $f(x) = 3x - 1$ then $f(4) = 3 \\times 4 - 1 = 11$. A composite function applies one function and then feeds the result into another. The notation $fg(x)$ means do $g$ first, then $f$, so $fg(x) = f(g(x))$. The order matters and is easy to get wrong. If $f(x) = 2x + 1$ and $g(x) = x^{2}$, then $fg(x) = 2x^{2} + 1$, but $gf(x) = (2x + 1)^{2}$, which is different. The inverse function $f^{-1}(x)$ reverses what $f$ does: it takes an output back to its input. To find it, write $y = f(x)$, swap the roles by making $x$ the subject, then rename. For $f(x) = 3x - 1$, set $y = 3x - 1$, rearrange to $x = \\frac{y + 1}{3}$, so $f^{-1}(x) = \\frac{x + 1}{3}$. A neat check is that $f(f^{-1}(x)) = x$, since applying a function and then its inverse returns the original input.

Graphs, gradient of a curve, and variation

You must recognise standard graph shapes. A linear graph $y = mx + c$ is a straight line with gradient $m$ and y-intercept $c$. A quadratic $y = ax^{2} + bx + c$ is a parabola, U-shaped when $a \\gt 0$ and n-shaped when $a \\lt 0$, with a single turning point. A cubic such as $y = x^{3}$ has a characteristic S-shape with up to two turning points. A reciprocal graph $y = \\frac{k}{x}$ is a hyperbola in two separate branches that never touch the axes, which act as asymptotes. An exponential graph $y = k a^{x}$ rises or falls steeply and approaches the x-axis but never reaches it. The gradient of a straight line is $\\frac{\\text{change in } y}{\\text{change in } x}$. For a curve the gradient changes from point to point, so to estimate it at a particular point you draw a tangent, a straight line just touching the curve there, and find the gradient of that tangent using two points read from it. Variation describes how quantities change together. Direct proportion is written $y \\propto x$, meaning $y = kx$ for a constant $k$; doubling $x$ doubles $y$, and the graph is a straight line through the origin. Inverse proportion is $y \\propto \\frac{1}{x}$, meaning $y = \\frac{k}{x}$; doubling $x$ halves $y$. To solve a variation problem, use one given pair of values to find $k$, then use that equation for any other case. Proportion can also involve squares or roots, such as $y \\propto x^{2}$ or $y \\propto \\frac{1}{\\sqrt{x}}$.