GCSE Higher Tier Algebra Reference Cheat Sheet

Key Facts

• Expanding brackets means multiplying every term, so $a(b + c) = ab + ac$ and $(x + a)(x + b) = x^2 + (a + b)x + ab$.
• A quadratic in standard form is $ax^2 + bx + c = 0$, where $a \neq 0$.
• The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for solving $ax^2 + bx + c = 0$.
• The discriminant $b^2 - 4ac$ tells the number of real roots: two if $b^2 - 4ac > 0$, one repeated if $b^2 - 4ac = 0$, and none if $b^2 - 4ac < 0$.
• Completing the square rewrites a quadratic as $a(x + p)^2 + q$, which shows the turning point at $(-p, q)$.
• When solving an inequality, multiplying or dividing by a negative number reverses the sign, so if $-2x < 6$, then $x > -3$.
• For an arithmetic sequence, the $n$th term is $u_n = a + (n - 1)d$, where $a$ is the first term and $d$ is the common difference.
• Function notation means an input is substituted into a rule, so if $f(x) = 2x - 3$, then $f(5) = 7$.

Vocabulary

Expression

An expression is a combination of numbers, variables, and operations without an equals sign, such as $3x^2 - 5x + 1$.

Equation

An equation is a mathematical statement with an equals sign that can be solved to find unknown values, such as $2x + 3 = 11$.

Factorise

To factorise means to rewrite an expression as a product of factors, such as $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Quadratic

A quadratic is an expression, equation, or function whose highest power of the variable is $2$, such as $y = x^2 - 4x + 1$.

Inequality

An inequality compares values using symbols such as $<$, $>$, $\leq$, or $\geq$ instead of an equals sign.

Function

A function is a rule that maps each input to exactly one output, often written using notation such as $f(x)$.