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Edexcel International AS Maths: Your notes

Pure 1
1.2 Quadratics
Contents
1.2.1 Quadratic Graphs
1.2.2 Discriminants
1.2.3 Completing the square
1.2.4 Solving Quadratic Equations
1.2.5 Further Solving Quadratic Equations (Hidden Quadratics)

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1.2.1 Quadratic Graphs

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Quadratic Graphs
What are quadratic graphs?
The general equation of a quadratic graph is y = ax 2 + bx + c
Their shape is called a parabola ("U" shape)
Positive quadratics have a value of a > 0 so the parabola is upright ∪
Negative quadratics have a value of a < 0 so the parabola is upside down ∩

Using quadratic graphs

You need to be able to:
sketch a quadratic graph given an equation or information about the graph
determine, from the equation, the axes intercepts
factorise, if possible, to find the roots of the quadratic function
find the coordinates of the turning point (maximum or minimum)
You may have to rearrange the equation before you can find some of these things

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Your notes

Exam Tip
Your calculator may tell you the roots of a quadratic function and the coordinates of the turning
point
But don't rely on it – think about how many marks the question is worth and how much
method/working you should show
Remember sometimes you'll need to rearrange an equation into the form

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1.2.2 Discriminants
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Discriminants
What is a discriminant?
The discriminant is the part of the quadratic formula that is under the square root sign b 2 − 4 ac
It is sometimes denoted by the Greek letter Δ (capital delta)
How does the discriminant affect graphs and roots?
There are three options for the outcome of the discriminant:

If b 2 − 4 ac > 0 the quadratic crosses the x-axis twice meaning there are two distinct real roots
If b 2 − 4 ac = 0 the quadratic touches the x-axis once meaning there is one real root (also called
repeated roots)
If b 2 − 4 ac < 0 the quadratic does not cross the x-axis meaning there are no real roots

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Discriminant and inequalities

You need to be able to set up and solve equations and inequalities (often quadratic) arising from the
discriminant

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Sketch the quadratic and decide whether you're looking above or below zero to write your solutions
correctly
Your notes

Exam Tip
When questions just mention “real roots”, the roots could be distinct or repeated (i.e. they arent
talking about complex numbers!)
In these cases, you only need to worry about solving
When solving using inequalities always sketch the quadratic and decide whether you're looking
above or below zero to help write your solutions correctly

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1.2.3 Completing the square

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Completing the square
What is completing the square?
Completing the square is another method used to solve quadratic equations
It simply means writing y = ax 2 + bx + c in the form y = a (x + p ) 2 + q
It can be used to help find other information about the quadratic like coordinates of the turning point
How do I complete the square?
The method used will depend on the value of the coefficient of the x2 term in y = ax 2 + bx + c

When a = 1
p is half of the coefficient of b
q is c - p2

When a ≠ 1

You first need to take a out as a factor of the x2 and x terms

Then continue as above

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When is completing the square useful?

Completing the square helps us find the turning point on a quadratic graph
It can also help you create the equation of a quadratic when given the turning point

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It can also be used to prove and/or show results using the fact that a squared term will always be
greater than or equal to 0

Exam Tip
Sometimes the question will explicitly ask you to complete the square
Sometimes it will even remind you of the form to write it in
But sometimes it will expect you to spot that completing the square is what you need to do to help
with other parts of the question... like finding turning points!

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1.2.4 Solving Quadratic Equations

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Solving Quadratic Equations
Solving quadratic equations
We can solve quadratic equations when they are written in the form ax 2 + bx + c = 0
If given an unusual looking equation, try to rearrange it into this form first
The three ways to solve a quadratic you must know are
Factorising
Completing the square
Quadratic formula
Solving a quadratic equation by factorising
Factorising is a great way to solve a quadratic quickly but won't work for all quadratics
If the numbers are simple, try factorising first
Once factorised, set each bracket to = 0 and solve

Solving a quadratic equation by completing the square

Completing the square will work for any quadratic
Make sure you know how to complete the square
Remember this will help with questions involving turning points too

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Solving a quadratic equation by the quadratic formula

The quadratic formula might look complicated but it just uses the coefficients a, b and c from the
quadratic equation
The quadratic formula will work for any quadratic

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Solving a quadratic equation by using a calculator

Most calculators now have the ability to solve quadratics

Get used to how your calculator functions work

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Your notes

So the solution to the quadratic equation 2x 2 + 5x − 12 = 0 are x = 1.5 and x = -4

Exam Tip
A calculator can be super-efficient but be aware some marks are for method
There will never be many marks for solving a quadratic at AS/A level
Use your judgement:
is it a “show that” or “prove” question?
how many marks?
how long is the question?
Remember the quadratic formula with a song... there are loads of fun ones on YouTube

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1.2.5 Further Solving Quadratic Equations (Hidden Quadratics)

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Further Solving Quadratic Equations (Hidden Quadratics)
What are hidden quadratic equations?
Hidden quadratic equations are quadratics written in terms of a function f (x )
A normal quadratic appears in the form ax 2 + bx + c = 0
Whereas a hidden quadratic appears in the form a ⎢⎣ f (x ) ⎥⎦ 2 + b ⎢⎣ f (x ) ⎥⎦ + c = 0
⎡ ⎤ ⎡ ⎤
This might look complicated but it simply means x has been replaced by f (x )
e.g. sin2 x + 2sin x − 3 = 0 is just the hidden quadratic of x 2 + 2x − 3 = 0 where
f (x ) = sin x
How to solve hidden quadratic equations
First rearrange the function into the form a ⎢⎣ f (x ) ⎥⎦ 2 + b ⎢⎣ f (x ) ⎥⎦ + c = 0
⎡ ⎤ ⎡ ⎤
Replacing the function and solving the 'normal' quadratic first
Then substitute the function back into the solutions to solve the original quadratic

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