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26 Dec 2025

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2 pages

Understanding Mathematical Expressions

B€LA〄

@bxla

Algebraic expressions are everywhere in maths, and mastering them is... Show more

Key words and definitions
Expression -Mathematical expression that consists of
variables, numbers and operations. Eg 4a + 2y
Term- A term is

Understanding Expressions and Key Techniques

Think of algebraic expressions like mathematical recipes - they contain variables (letters), numbers, and operations all mixed together. Each part separated by + or - signs is called a term, and you'll often need to rearrange or rewrite these expressions in different forms.

Expanding single brackets is straightforward - multiply everything inside the bracket by the number outside. For example, 2 $x+8$ becomes 2x + 16. When you see double brackets like $x+8$ $x+5$ , use the FOIL method: multiply First terms, Outer terms, Inner terms, then Last terms.

Factorising is the reverse process - you're putting expressions back into brackets. Start by looking for the highest common factor (HCF) of all terms. For instance, 5x + 25 becomes 5 $x + 5$ because 5 is the HCF of both terms.

Top tip: Always check your factorising by expanding it back out - you should get your original expression!

The difference of two squares follows a special pattern: a² - b² = $a-b$ $a+b$ . So x² - 81 becomes $x-9$ $x+9$ because 81 = 9².

Mastering Quadratic Expressions and Equations

Quadratic expressions (those with x² terms) might look intimidating, but they follow predictable patterns. To factorise x² - 3x + 2, you need two numbers that multiply to give +2 and add to give -3. That's -1 and -2, so the answer is $x-1$ $x-2$ .

When factorising quadratics, pay close attention to the signs. If the constant term is positive, both factors have the same sign as the middle term. If it's negative, the factors have opposite signs.

Solving quadratic equations by factorising uses a clever trick: if two things multiply to give zero, one of them must be zero. So if $x+7$ $x-2$ = 0, then either x+7 = 0 or x-2 = 0, giving you x = -7 or x = 2.

Remember: Quadratic equations usually have two solutions, so don't stop after finding the first one!

Practice makes perfect with these techniques. Start with simpler expressions and gradually work up to more complex ones. Once you've mastered the patterns, you'll spot the solutions much more quickly.

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415

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26 Dec 2025

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2 pages

Understanding Mathematical Expressions

B€LA〄

@bxla

Algebraic expressions are everywhere in maths, and mastering them is your key to unlocking higher-level topics. This guide breaks down the essential skills you need: expanding brackets, factorising expressions, and solving quadratic equations through clear steps and examples.

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Understanding Expressions and Key Techniques

Top tip: Always check your factorising by expanding it back out - you should get your original expression!

The difference of two squares follows a special pattern: a² - b² = $a-b$ $a+b$ . So x² - 81 becomes $x-9$ $x+9$ because 81 = 9².

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Mastering Quadratic Expressions and Equations

When factorising quadratics, pay close attention to the signs. If the constant term is positive, both factors have the same sign as the middle term. If it's negative, the factors have opposite signs.

Remember: Quadratic equations usually have two solutions, so don't stop after finding the first one!