International Baccalaureate Mathematics

Syllabus Content

Solution of quadratic equations and inequalities. The quadratic formula. The discriminant ∆=b²-4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Factorising Video A reminder of how to factorise an algebraic expression. This video is to help you do the online, self-marking exercise.

New Way to Solve Quadratics A computationally-efficient, natural, and easy-to-remember algorithm for solving general quadratic equations.

Quadratic Equations Video Learn the common methods of solving quadratic equations by factorising and by using the quadratic formula.

Quadratic Formula Song A song from Math Upgrade dot com.

Quadratic Equations Solve these quadratic equations algebraically in this seven-level, self-marking online exercise.

Factorising Quadratics An exercise about factorising quadratics presented one question at a time suitable for a whole class activity.

Completing the Square Practise this technique for solving quadratic equations and analysing graphs.

Name that Graph Choose from the given functions the one that best describes the shape of the graph.

Here are some exam-style questions on this statement:

"The diagram below is a sketch of $y = f(x)$ where $f(x)$ is a quadratic function." ... more

"The volume of the larger red cube is 386cm³ greater than the volume of the smaller blue cube." ... more

"(a) Show that the equation $\frac{3}{x+1}+\frac{3x-9}{2}=1$ can be simplified to $3x^2-8x-5=0$." ... more

"A red rug has a width of $x-3$ cm and a length of $4x$ cm." ... more

"In the diagram below, which is not drawn to scale, all dimensions are in centimetres and all angles are multiples of 90^o. If the shaded area is 698cm², work out the value of $x$." ... more

"Given that:" ... more

"The area of triangle ABC (not drawn to scale) is " ... more

See all these questions

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Algebra Pupils begin their study of algebra by investigating number patterns. Later they construct and express in symbolic form and use simple formulae involving one or many operations. They use brackets, indices and other constructs to apply algebra to real word problems. This leads to using algebra as an invaluable tool for solving problems, modelling situations and investigating ideas. If this topic were split into four sub topics they might be: Creating and simplifying expressions; Expanding and factorising expressions; Substituting and using formulae; Solving equations and real life problems; This is a powerful topic and has strong links to other branches of mathematics such as number, geometry and statistics. See also "Number Patterns", "Negative Numbers" and "Simultaneous Equations".

Furthermore

Official Guidance, clarification and syllabus links:

Using factorization, completing the square (vertex form), and the quadratic formula.

Solutions may be referred to as roots or zeros.

Example: For the equation $3kx^2+2x+k=0$, find the possible values of $k$, which will give two distinct real roots, two equal real roots or no real roots.

Formula Booklet:

Solutions of a quadratic equation	$ax^2 + bx + c = 0 \implies x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} $
Discriminant	$\Delta = b^2 - 4ac $

The solution of quadratic equations is a fundamental concept in algebra. A quadratic equation is typically in the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are coefficients. The solutions to these equations can be found using the quadratic formula:

$$ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} $$

This formula calculates the roots of the quadratic equation. The nature of these roots is determined by the discriminant $ \Delta = b^2 - 4ac $. The discriminant reveals:

Two distinct real roots if $ \Delta > 0 $
Two equal real roots (or one real root) if $ \Delta = 0 $
No real roots if $ \Delta < 0 $

Example: Consider the quadratic equation $ 2x^2 - 4x - 6 = 0 $. To find the roots:

Identify the coefficients: $ a = 2 $, $ b = -4 $, and $ c = -6 $.
Compute the discriminant: $ \Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64 $.
Since $ \Delta > 0 $, there are two distinct real roots.
Apply the quadratic formula:

$$ x = \frac{{-(-4) \pm \sqrt{{64}}}}{{2 \times 2}} = \frac{{4 \pm 8}}{4} $$

Hence, the solutions are $ x = 3 $ and $ x = -1 $.

This video on Factorising Quadratic Functions and Equations is from Revision Village and is aimed at students taking the IB Maths AA Standard level course.

This video on Discriminant Test (Quadratics) is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

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Solutions of a quadratic equation	\(ax^2 + bx + c = 0 \implies x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \)
Discriminant	\(\Delta = b^2 - 4ac \)