The quadratic formula is a powerful tool for solving quadratic...
Maths240 views·Updated 19 Aug 2026·5 pagesMaster the Nat 5 Maths Quadratic Formula
A
Amilie du Toit@amiliedutoit_uajk
1 / 5
1
of 5
Using the Quadratic Formula
The quadratic formula is: where a is the coefficient of x², b is the coefficient of x, and c is the constant term.
Let's see how it works with an example: . We identify a=1, b=6, and c=2, then substitute into the formula. After calculating the discriminant (), we get two solutions: or .
For another example, , we have a=3, b=-10, and c=2. Working through the formula gives us or .
Remember: The ± sign in the formula means you'll always get two possible solutions (unless the discriminant equals zero, which gives one solution).
2
of 5
Solving Word Problems with Quadratics
Real-world problems often require quadratic equations. Let's tackle one: a rectangular playground is 10m longer than it is wide with an area of 1400m².
To solve this, we need to set up a quadratic equation. If the width is w, then the length is . The area formula gives us: 1400 = w+10$$w, which expands to w² + 10w - 1400 = 0.
Using the quadratic formula with a=1, b=10, c=-1400: . This gives us w ≈ 32.75m (we reject the negative solution as width can't be negative). Therefore, the length is 32.75 + 10 = 42.75m.
Pro tip: Always check if your answer makes sense in the context of the problem. For example, measurements like width can't be negative!
3
of 5
Finding Quadratic Equations from Points
You can determine the equation of a quadratic function if you know some points on the graph. For the form y = kx², you only need one non-zero point.
For example, if the graph passes through (0,0) and (1,3), we can find k by substituting the second point: 3 = k(1)², so k = 3. The equation is y = 3x².
Similarly, for a graph through (0,0) and , we substitute to get -8 = k(2)², so k = -2. The equation becomes y = -2x².
Quick check: You can verify your equation by testing it with the original points. If y = 3x² and x = 1, then y should equal 3.
4
of 5
Understanding the Form y = ² + q
The form y = ² + q helps us understand how a parabola is positioned on a graph. The value p shifts the graph horizontally, and q shifts it vertically.
If we know that p = 2 and q = 3, the equation is y = ² + 3. This means the parabola is shifted 2 units right and 3 units up from the standard position.
For a graph where p = -2 and q = -1, the equation becomes y = ² - 1. This parabola is shifted 2 units left and 1 unit down from the standard position.
Visual aid: Think of p and q as giving the coordinates of the turning point (p,q) - this is where the parabola reaches its minimum or maximum value.
5
of 5
Finding the Equation of a Parabola h
When given specific information about a parabola in the form h = ² + q, we can determine its equation.
If a parabola's axis of symmetry is at x = -2 and it passes through (3,0) and , then p = -2 (the axis of symmetry). To find q, we substitute one of the points: 0 = ² + q, which gives us 0 = 25 + q, so q = -25.
However, we should verify with the other point: 0 = ² + q means 0 = 1 + q, so q = -1. Since we get different values for q, we need to double-check our working.
Important: The axis of symmetry for a parabola in the form ² + q is always x = p, which is midway between the x-intercepts if they exist.
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Maths240 views·Updated 19 Aug 2026·5 pagesMaster the Nat 5 Maths Quadratic Formula
A
Amilie du Toit@amiliedutoit_uajk
The quadratic formula is a powerful tool for solving quadratic equations when they can't be easily factored. This formula works for any quadratic equation in the form ax² + bx + c = 0, giving you the exact values where...
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of 5
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Using the Quadratic Formula
The quadratic formula is: where a is the coefficient of x², b is the coefficient of x, and c is the constant term.
Let's see how it works with an example: . We identify a=1, b=6, and c=2, then substitute into the formula. After calculating the discriminant (), we get two solutions: or .
For another example, , we have a=3, b=-10, and c=2. Working through the formula gives us or .
Remember: The ± sign in the formula means you'll always get two possible solutions (unless the discriminant equals zero, which gives one solution).
2
of 5Sign up to see the content. It's free!
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Solving Word Problems with Quadratics
Real-world problems often require quadratic equations. Let's tackle one: a rectangular playground is 10m longer than it is wide with an area of 1400m².
To solve this, we need to set up a quadratic equation. If the width is w, then the length is . The area formula gives us: 1400 = w+10$$w, which expands to w² + 10w - 1400 = 0.
Using the quadratic formula with a=1, b=10, c=-1400: . This gives us w ≈ 32.75m (we reject the negative solution as width can't be negative). Therefore, the length is 32.75 + 10 = 42.75m.
Pro tip: Always check if your answer makes sense in the context of the problem. For example, measurements like width can't be negative!
3
of 5Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding Quadratic Equations from Points
You can determine the equation of a quadratic function if you know some points on the graph. For the form y = kx², you only need one non-zero point.
For example, if the graph passes through (0,0) and (1,3), we can find k by substituting the second point: 3 = k(1)², so k = 3. The equation is y = 3x².
Similarly, for a graph through (0,0) and , we substitute to get -8 = k(2)², so k = -2. The equation becomes y = -2x².
Quick check: You can verify your equation by testing it with the original points. If y = 3x² and x = 1, then y should equal 3.
4
of 5Sign up to see the content. It's free!
- Access to all documents
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By signing up you accept Terms of Service and Privacy Policy
Understanding the Form y = ² + q
The form y = ² + q helps us understand how a parabola is positioned on a graph. The value p shifts the graph horizontally, and q shifts it vertically.
If we know that p = 2 and q = 3, the equation is y = ² + 3. This means the parabola is shifted 2 units right and 3 units up from the standard position.
For a graph where p = -2 and q = -1, the equation becomes y = ² - 1. This parabola is shifted 2 units left and 1 unit down from the standard position.
Visual aid: Think of p and q as giving the coordinates of the turning point (p,q) - this is where the parabola reaches its minimum or maximum value.
5
of 5Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding the Equation of a Parabola h
When given specific information about a parabola in the form h = ² + q, we can determine its equation.
If a parabola's axis of symmetry is at x = -2 and it passes through (3,0) and , then p = -2 (the axis of symmetry). To find q, we substitute one of the points: 0 = ² + q, which gives us 0 = 25 + q, so q = -25.
However, we should verify with the other point: 0 = ² + q means 0 = 1 + q, so q = -1. Since we get different values for q, we need to double-check our working.
Important: The axis of symmetry for a parabola in the form ² + q is always x = p, which is midway between the x-intercepts if they exist.
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1080,2436,325
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Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user