•

Updated Mar 18, 2026

•

7 pages

Junior Cycle Maths: Mastering Algebra

Kyla McInerney

@kylamcinerney

Algebra might seem intimidating at first, but it's actually just... Show more

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# Junior Cort

AIGEBRA.

(1st year) Ruies of algebra.

I adding & Subtraching.

Same sigo Keep the scale Signo add

diff Sign Keep Sign of b

First Year Algebra Rules

Adding and subtracting with positive and negative numbers follows two simple rules. When the signs are the same, keep that sign and add the numbers together. When the signs are different, keep the sign of the bigger number and subtract the smaller from the larger.

Multiplying and dividing works similarly - same signs give positive answers, different signs give negative answers. It's just like normal multiplication, but you need to watch those signs carefully.

When multiplying with letters (like x or y), multiply the numbers first, then the letters. The key trick is when you multiply the same letters together, you add their powers - so x × x² = x³.

Quick Tip: Think of algebra like a balance scale - whatever you do to one side, you must do to the other to keep it balanced!

Solving equations means finding what value makes the equation true. Always keep both sides balanced by doing the same operation to each side until you isolate the variable.

Second Year Factorising

Factorising is like reverse multiplication - you're breaking expressions down into smaller parts that multiply together. Start by looking for the Highest Common Factor (HCF), which is the biggest number or letter that divides into all terms.

Difference of two squares follows a special pattern: a² - b² = $a + b$ $a - b$ . This only works when you're subtracting one perfect square from another, like x² - 9 = $x + 3$ $x - 1$ .

Grouping helps when you have four terms that can be paired up. Look for common factors in pairs of terms, then see if you can factor out a common bracket.

Memory Hook: Always check your factorising by expanding back out - you should get your original expression!

Quadratic expressions can sometimes be factorised into two brackets. Look for two numbers that multiply to give the constant term and add to give the middle coefficient.

Simultaneous and Quadratic Equations

Simultaneous equations involve two equations with two unknowns that you solve together. The elimination method works by making one variable disappear when you add or subtract the equations.

Word problems become much easier when you define your variables clearly first. Always read the problem twice and identify what each variable represents before writing your equations.

Quadratic equations can be solved using the quadratic formula: x = $-b ± √(b² - 4ac)$ /2a. This works for any quadratic equation written as ax² + bx + c = 0.

Exam Tip: Pay attention to how many decimal places the question asks for - this tells you whether to use the formula or try factorising first!

Sometimes you can solve quadratics by factorising them first, which is often quicker than using the formula. If it factors into $x + p$ $x + q$ = 0, then x = -p or x = -q.

More Quadratic Methods

Factorising quadratics often starts with taking out common factors first. Look for HCF before trying other methods - it makes everything simpler.

The difference of two squares pattern appears frequently in quadratics. Remember that a² - b² always factors as $a + b$ $a - b$ , which gives you two solutions immediately.

For more complex quadratics, you might need to use grouping or the quadratic formula. Don't be afraid to try factorising first - if it doesn't work easily, switch to the formula.

Success Strategy: Most quadratic equations in exams are designed to factor nicely, so always try this method first!

When solving quadratic equations, remember you're looking for two solutions in most cases. Check both answers make sense in the original problem, especially for word problems.

Forming Quadratic Equations

Sometimes you'll need to work backwards from the solutions to create the quadratic equation. If you know the roots are p and q, then the equation is $x - p$ $x - q$ = 0.

The process is straightforward: write each root as x = p, rearrange to x - p = 0, then multiply the brackets together. This gives you the quadratic in standard form.

Expanding the brackets carefully is crucial here. Use the FOIL method (First, Outside, Inside, Last) or the grid method to make sure you don't miss any terms.

Check Your Work: Substitute your original roots back into your final equation - you should get zero each time!

This skill is particularly useful for reverse engineering problems where you need to find an equation with specific properties or solutions.

Algebraic Fractions - Adding and Subtracting

Adding algebraic fractions works just like adding regular fractions - find a common denominator first. When denominators are numbers, find the LCD (Lowest Common Denominator) as usual.

The tricky part comes when you have algebra in the denominators. Here, your LCD becomes the product of all the different factors in the denominators.

Always multiply both numerator and denominator by whatever makes the denominators match. This keeps the fraction equivalent while giving you a common base to work from.

Pro Tip: Write out each step clearly - algebraic fractions have lots of moving parts, and small mistakes add up quickly!

When subtracting fractions, be extra careful with negative signs. Distribute the minus sign through the entire numerator of the fraction you're subtracting.

Division in Algebra

Simplifying algebraic fractions means cancelling common factors from the numerator and denominator. Look for HCF first, then check if either part can be factorised further.

Quadratics in fractions often need factorising before you can simplify. Factor both the numerator and denominator separately, then cancel any common factors.

Long division in algebra follows the same pattern as with numbers, but you're dividing terms with variables. Divide the leading terms first, multiply back through, subtract, and bring down the next term.

Division Check: Multiply your answer by the divisor - you should get back to your original expression!

The key to success with algebraic division is working systematically and checking each step. Don't rush - these problems reward careful, methodical work.

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Mathematics

•

Updated Mar 18, 2026

•

7 pages

Junior Cycle Maths: Mastering Algebra

Kyla McInerney

@kylamcinerney

Algebra might seem intimidating at first, but it's actually just like solving puzzles with numbers and letters! These notes cover everything from basic rules in first year through advanced techniques in third year, giving you all the tools you need... Show more

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First Year Algebra Rules

When multiplying with letters (like x or y), multiply the numbers first, then the letters. The key trick is when you multiply the same letters together, you add their powers - so x × x² = x³.

Quick Tip: Think of algebra like a balance scale - whatever you do to one side, you must do to the other to keep it balanced!

Solving equations means finding what value makes the equation true. Always keep both sides balanced by doing the same operation to each side until you isolate the variable.

Sign up to see the contentIt's free!

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Improve your grades

Join milions of students

Second Year Factorising

Difference of two squares follows a special pattern: a² - b² = $a + b$ $a - b$ . This only works when you're subtracting one perfect square from another, like x² - 9 = $x + 3$ $x - 1$ .

Grouping helps when you have four terms that can be paired up. Look for common factors in pairs of terms, then see if you can factor out a common bracket.

Memory Hook: Always check your factorising by expanding back out - you should get your original expression!

Quadratic expressions can sometimes be factorised into two brackets. Look for two numbers that multiply to give the constant term and add to give the middle coefficient.

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Simultaneous and Quadratic Equations

Simultaneous equations involve two equations with two unknowns that you solve together. The elimination method works by making one variable disappear when you add or subtract the equations.

Word problems become much easier when you define your variables clearly first. Always read the problem twice and identify what each variable represents before writing your equations.

Quadratic equations can be solved using the quadratic formula: x = $-b ± √(b² - 4ac)$ /2a. This works for any quadratic equation written as ax² + bx + c = 0.

Exam Tip: Pay attention to how many decimal places the question asks for - this tells you whether to use the formula or try factorising first!

Sometimes you can solve quadratics by factorising them first, which is often quicker than using the formula. If it factors into $x + p$ $x + q$ = 0, then x = -p or x = -q.

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More Quadratic Methods

Factorising quadratics often starts with taking out common factors first. Look for HCF before trying other methods - it makes everything simpler.

The difference of two squares pattern appears frequently in quadratics. Remember that a² - b² always factors as $a + b$ $a - b$ , which gives you two solutions immediately.

For more complex quadratics, you might need to use grouping or the quadratic formula. Don't be afraid to try factorising first - if it doesn't work easily, switch to the formula.

Success Strategy: Most quadratic equations in exams are designed to factor nicely, so always try this method first!

When solving quadratic equations, remember you're looking for two solutions in most cases. Check both answers make sense in the original problem, especially for word problems.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Forming Quadratic Equations

Sometimes you'll need to work backwards from the solutions to create the quadratic equation. If you know the roots are p and q, then the equation is $x - p$ $x - q$ = 0.

The process is straightforward: write each root as x = p, rearrange to x - p = 0, then multiply the brackets together. This gives you the quadratic in standard form.

Expanding the brackets carefully is crucial here. Use the FOIL method (First, Outside, Inside, Last) or the grid method to make sure you don't miss any terms.

Check Your Work: Substitute your original roots back into your final equation - you should get zero each time!

This skill is particularly useful for reverse engineering problems where you need to find an equation with specific properties or solutions.

Sign up to see the contentIt's free!

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Improve your grades

Join milions of students

Algebraic Fractions - Adding and Subtracting

Adding algebraic fractions works just like adding regular fractions - find a common denominator first. When denominators are numbers, find the LCD (Lowest Common Denominator) as usual.

The tricky part comes when you have algebra in the denominators. Here, your LCD becomes the product of all the different factors in the denominators.

Always multiply both numerator and denominator by whatever makes the denominators match. This keeps the fraction equivalent while giving you a common base to work from.

Pro Tip: Write out each step clearly - algebraic fractions have lots of moving parts, and small mistakes add up quickly!

When subtracting fractions, be extra careful with negative signs. Distribute the minus sign through the entire numerator of the fraction you're subtracting.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Division in Algebra

Simplifying algebraic fractions means cancelling common factors from the numerator and denominator. Look for HCF first, then check if either part can be factorised further.

Quadratics in fractions often need factorising before you can simplify. Factor both the numerator and denominator separately, then cancel any common factors.

Division Check: Multiply your answer by the divisor - you should get back to your original expression!

The key to success with algebraic division is working systematically and checking each step. Don't rush - these problems reward careful, methodical work.