Tricky topics in GCSE Maths: Quadratic Equations
Through analysing the usage data of Tassomai students and identifying some common mistakes in their quizzes we can see which topics GCSE maths students struggle with the most. In this series of blogs we’ll post a brief explainer on each of these tricky topics to help GCSE maths students get up to speed and prepare for exams.
Quadratic Equations is an important topic for GCSE maths students to get to grips with, as it’s an exam specification point for major exam boards including:
✔ AQA
✔ EDEXCEL
✔ OCR 21ST CENTURY
✔ OCR GATEWAY
✔ WJEC
What is a Quadratic Equation?
A quadratic equation is a type of polynomial equation of the second degree. The general form of a quadratic equation is:
\[ax^2 + bx + c = 0\]
Here:
\(x\) is the variable or unknown we want to solve for.
\(a\), \(b\), and \(c\) are constants (numbers that don’t change).
\(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.
Key Characteristics
Degree: This is the highest power of \(x\) in an equation, which here is 2.
Shape: When graphed, a quadratic equation forms a parabola (a u-shaped curve), which can either open upwards (if \(a>0\)) or downwards (if \(a<0\)).
Here’s a Tassomai quiz question about Quadratic Equations:
This is one of several retrieval practice questions about Quadratic Equations that’s shown to Tassomai students. Retrieval practice is a proven learning method to embed knowledge through testing.
The question asks you:
TRUE or FALSE?: If you can FACTORISE a QUADRATIC equation, you can easily determine the ROOTS (the solutions), e.g. \[(x+3)(x-7)=0\]
1. [TRUE]: The two brackets tell you the answer - you just use the numbers you see. Here, \(x=3\) or \(x=-7\).
2. [TRUE]: The two brackets multiply to make ZERO, so one or the other must equal zero. If \((x+3)=0\), then \(x=-3\); if \((x-7)=0\), then \(x=7\).
3. FALSE: The solution to the equation is visible BEFORE you factorise: in this case \(x^2-4x-21\) gives us the root of \(x=\pm21\)
4. FALSE: You cannot solve a quadratic from these methods
To help us dive deeper into this topic we’ve enlisted the help of Mai, Tassomai’s built-in AI-powered tutor.
Tassomai users can ask Mai for help if they’re stuck on a quiz question or need some terminology explained to them.
If a student chooses the wrong answer, they can ask Mai to explain why it’s incorrect.
Let’s ask Mai to explain the answers to the quiz question!
Understanding Quadratic Roots Through Factorisation
When you factorise a quadratic equation, you’re setting the stage to find its roots easily. Let’s break down why the correct answer is the best choice and where the other options miss the mark:
The Correct Answer: True, Factorisation Reveals the Roots
Zero Product Property: If the product of two factors is zero, at least one of the factors must be zero. This principle lets us solve \((x+3)(x-7)=0\) by setting each bracket equal to zero.
Finding Roots: Solving \((x+3)=0\) gives \(x=−3\), and \((x−7)=0\) gives \(x=7\). These are the roots of the equation, showing where the graph crosses the x-axis.
Why the Other Options Don’t Fit:
Misinterpretation of Signs: The statement “The two brackets tell you the answer - you just use the numbers you see. Here, \(x=3\) or \(x=−7\) misunderstands how factorisation works. The signs in the factors are crucial; they must be considered as they appear directly in the factors.
“The solution to the equation is visible BEFORE you factorise: in this case \(x^2−4x−21\) gives us the root of \(x=\pm21\): This is incorrect because the roots \(x=\pm21\) don’t actually solve the original quadratic equation. Proper factorisation and solving are necessary to find the correct roots.
“You cannot solve a quadratic from these methods”: This is also false. Factorisation is a standard and effective method to solve quadratic equations, especially when they can be expressed as the product of two linear factors.
Fun Fact:
Did you know that the method of solving quadratic equations by factorisation dates back to ancient mathematicians? They knew a thing or two about numbers!
By understanding the correct process and keeping an eye on the signs, you’ll be a factorising wizard in no time! 🧙♂️✨
Here’s how Mai’s explanation would look when shown to students using the Tassomai app.
As you can see, Mai sometimes throws in a fun fact to help students remember a topic.
Learn more about Quadratic Equations
We hope you found Mai’s explanation helpful. If you’d like to learn more about Quadratic Equations this GCSE maths Live Lesson presented by Talib (a teacher that uses Tassomai in his school) which focuses on Algebra.
The key content about spelling and grammar starts at around 6 minutes into the recording.
To see more tricky GCSE topics explained, click here for the full list of Tricky Topics blogs.
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