Factorising Quadratic Expressions Revision Notes for Leaving Cert Mathematics

Factorising Quadratic Expressions

What is a quadratic expression?

A quadratic expression is any mathematical expression that can be written in the form $ax^2 + bx + c$ , where $a$ , $b$ and $c$ are numbers and $a \neq 0$ . The key feature is that the highest power of the variable is 2.

Examples of quadratic expressions include:

$x^2 + 7x + 10$
$3x^2 + 13x + 4$
$2x^2 - 5x + 3$

When we factorise a quadratic expression, we rewrite it as a product of two or more simpler expressions.

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Factorisation Example

For example, $x^2 + 7x + 10$ can be factorised as $(x + 5)(x + 2)$ .

You can verify this by expanding: $(x + 5)(x + 2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10$ ✓

Method 1: Trial and error factorisation

This method works for quadratic expressions of the form $ax^2 + bx + c$ . We look for two brackets that, when multiplied together, give us back the original expression.

The process

When factorising using trial and error, we need to find numbers such that:

The product of the outside terms added to the product of the inside terms gives the middle term of the quadratic expression
The product of the first terms gives the $x^2$ term
The product of the last terms gives the constant term

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Worked Example: Factorise $3x^2 + 13x + 4$

Step 1: The factors will take the form $(3x + ?)(x + ?)$

Step 2: We need to find factors of 4 that work. The factors of 4 are: $4 \times 1$ or $2 \times 2$

Step 3: Try $(3x + 1)(x + 4)$ :

First terms: $3x \times x = 3x^2$ ✓
Outside terms: $3x \times 4 = 12x$
Inside terms: $1 \times x = x$
Last terms: $1 \times 4 = 4$ ✓
Middle term: $12x + x = 13x$ ✓

Answer: $3x^2 + 13x + 4 = (3x + 1)(x + 4)$

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Worked Example: Factorise $8x^2 + 10x - 3$

Step 1: The factors will take the form $(4x - 1)(2x + 3)$

Step 2: Check this works:

First terms: $4x \times 2x = 8x^2$ ✓
Outside terms: $4x \times 3 = 12x$
Inside terms: $-1 \times 2x = -2x$
Last terms: $-1 \times 3 = -3$ ✓
Middle term: $12x - 2x = 10x$ ✓

Answer: $8x^2 + 10x - 3 = (4x - 1)(2x + 3)$

Method 2: Taking out common factors

For expressions of the form $ax^2 + bx$ (where there's no constant term), we can factor out the common factor.

The process

Look for the highest common factor of all terms and take it outside the brackets.

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Worked Examples: Common Factor Method

$x^2 - 5x = x(x - 5)$
$3x^2 - 6x = 3x(x - 2)$
$9x^2 - 15x = 3x(3x - 5)$

The common factor is always taken out first, leaving a simpler expression in the brackets.

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Why This Method Works

When you have terms that share common factors, factoring them out simplifies the expression and often reveals further factorisation opportunities.

Method 3: Difference of two squares

Perfect squares are numbers like 1, 4, 9, 16, 25... which can be written as $1^2$ , $2^2$ , $3^2$ , $4^2$ , $5^2$ ...

Similarly, expressions like $9x^2$ and $16y^2$ are squares since $9x^2 = (3x)^2$ and $16y^2 = (4y)^2$ .

When we have an expression like $9x^2 - 16y^2$ , this is called the difference of two squares.

The formula

$x^2 - y^2 = (x + y)(x - y)$

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Critical Formula to Memorise

The difference of two squares formula $x^2 - y^2 = (x + y)(x - y)$ is a key formula you must memorise for the exam. This pattern appears frequently in examinations.

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Worked Example: Factorise $x^2 - 25$

This is $(x)^2 - (5)^2$

Using the formula: $x^2 - 25 = (x + 5)(x - 5)$

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Worked Example: Factorise $9x^2 - 16y^2$

This is $(3x)^2 - (4y)^2$

Using the formula: $9x^2 - 16y^2 = (3x + 4y)(3x - 4y)$

Practice problems

Here are some quadratic expressions you should be able to factorise using these methods:

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Practice Expression Types

Common types you'll encounter include:

Simple trial and error: $x^2 + 7x + 6$ , $x^2 - 7x + 12$
More complex coefficients: $2x^2 + 5x + 2$ , $3x^2 + 8x + 4$
Difference of squares: expressions like $x^2 - 25$ , $4x^2 - 9$
Common factors: $2x^2 + 6x$ , $5x^2 - 15x$

Exam tips and common mistakes

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Essential Exam Strategies

Always check your answer by expanding the brackets to see if you get back to the original expression
Look for common factors first before trying other methods - this often simplifies the problem significantly
Remember the difference of two squares pattern - it appears frequently in exams and is easily recognisable
Be careful with signs - negative terms can be tricky in the trial and error method
Practice identifying which method to use - this skill comes with experience and saves valuable exam time

Common Mistakes to Avoid:

Forgetting to check for common factors before factorising
Making sign errors when dealing with negative coefficients
Not recognising difference of squares patterns
Rushing the trial and error process without systematic checking

Remember!

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Key Points to Remember:

Quadratic expressions have the form $ax^2 + bx + c$ where $a \neq 0$
Trial and error works by finding factors that give the correct first, middle, and last terms when expanded
Common factors should always be taken out first when present - this simplifies the remaining factorisation
Difference of two squares follows the pattern $x^2 - y^2 = (x + y)(x - y)$
Always check your factorisation by expanding the brackets back out to verify your answer

Factorising Quadratic Expressions (Leaving Cert Mathematics): Revision Notes