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How to Revise: Square Form and Nature of Roots


Square Form and the Nature of Roots are a common feature of the LCHL Paper 1 (see list of appearances below).


NB: But that is NOT the main reason why students should put effort into revising these concepts.


These concepts are two of the more advanced applications of the Factor Theorem.

A student's mastery of these topics is a good indicator as to their understanding of the Factor Theorem.


Link to Module and Worksheet Preview

The Tacit Maths

Download a preview of the worksheet (which corresponds to eLearning Modules (for students) and Powerpoint (for teachers):


WORKSHEET WM Square Form & Nature of Roots
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Tips when Revising Square Form and the Nature of Roots

  1. Know the Factor Theorem

It may seem obvious, but it is pointless tackling these two concepts if you are not up to speed with the Factor theorem.

The 3 Key Features of a Quadratic Function are:

  1. It's Shape

  2. It's Roots

  3. The y-intercept

Work on understanding these, and the rest should follow.

(Note: The "Factor Theorem and Functions & Graphs" module should get you fully up to speed!)


Square Form

2. It's OK to Memorise...

Worksheet: Page 4 - 5

Powerpoint: Page 8 - 11

One aim of the LC Maths course is to discourage "rote learning" or "memorisation".

Square form, however, is one example where mastering it involves learning off the process step by step.

But that's OK! A deeper understanding of Square Form and its applications usually come once it is learned off.


3. Work with Decimals AND Fractions

A good indicator of a well prepared LCHL student is their ability to work interchangeably with fractions/decimals/scientific notation etc.

Square Form is a good example of where students often tend towards solely using decimals.

However, once you get into the habit of using fraction more often, it improves your understanding of the rules of





Nature of Roots:

4. What is a Quadratic anyway??

Worksheet: Page 13 - 14

Powerpoint: Page 62

One of the key hurdles that students need to climb in order to master the Nature of Roots is identifying a Quadratic expression.

This may seem obvious.

However, look at the exercise shown here.

Students need to understand:

(a) That each of these expressions are in fact Quadratic Expressions

(b) How to identify the values of "a", "b" and "c" for each expression.

NB: There is no point in continuing with the Nature of Roots if this isn't clear.


5. Know how to work with Inequalities

Very often, a student may have a good understanding of how to use the discriminant, but not have the skills to solve a Quadratic Inequality. This is a huge part of this topic.


Remember: If the solution requires solving a Quadratic Inequality, and you do not leave your answer in domain form, you could lose up to FOUR marks for that question.

(Note: The "Inequalities & Modules" module could be helpful here!)


6. How/When they appear in the Leaving Cert

At first glance, Square Form and the Nature of Roots seem to be completely separate concepts.

However, they have a lot in common, and as a result they often appear together in the Leaving Cert.

Here is a list of when and how Square Form and the Nature of Roots has appeared over the last 7 years of the Leaving Cert:

LC 2022: Sq. Form and Nature of Roots

LC 2021: Sq. Form only (with Calculus)

LC 2020: Nature of Roots (5 Mark Question)

LC 2019: N/A

LC 2018: N/A

LC 2017: Sq. Form and Nature of Roots (Full 25 Mark Question)

LC 2016: Sq. Form (5 Mark Question)

LC 2015: N/A

LC 2014: Full 25 Mark Question with a CUBIC Function (NOTE: HARD!)




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