AS LEVEL QUADRATIC (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

Mathematic pure 1 (Quadratics)email: racsostudenthelp@gmail.com
1. Quadratics
ax2 + bx + c = a(x +
b
2a
) 2 + d
Why do we change quadratic equations/ expressions into a completing square form?
- We could easily find the co-ordinates of the vertex( turning point)
- Having the coordinates of the vertex you could find the range and/or domain
of the graph.( Domain and Range will be explained in functions)
Consider the graph on the following page:
RACSO PRODUCTS Page 1

Y-axis
X-axis
Vertex (turning point)
How to write a quadratic equation/expression into the completing square form;
- I will explain it in 3 different cases, when “a” is +ve, when “a” is -ve and when
“a” is any number bigger that 1, example 2, 3 or 6.

- When “a” is (+1)
X2 - 4x + 7
a=(+1), b=(-4) and c=(+7)
Note: x2 also means 1x2
X2 - 4x + (b/2a)2 –(b/2a)2 + 7
X2 - 4x +(b/2a)2 + 7 –(b/2a)2
X2 - 4x + (-2)2 + 7 - (-2)2
(x-2)2 +3
=(x-2)2 +3
X2 - 4x + 7 = (x-2)2 +3
Quadratics expression
This expression is mathematically equal to the above
quadratic expression ( because I added (b/2a)2 and subtracted
(b/2a)2 )
You have (x2) and a number between, then (-2)2. Combining the
two you get ( x-2)2
This is the answer, now it is in completing square form.

When “a” is (-1)
-X2 - 4x + 7
Note: -x2 also means -1x2
-X2 - 4x + 7
- (x2 + 4x) + 7
-(x2 + 4x + (b/2a)2 –(b/2a)2)+ 7
-(x2 + 4x + (2)2 –(2)2) + 7
-(x2 + 4x + (2)2)+ 7+(2)2
-(x2 + 4x + (2)2)+ 7+(2)2
-(x+2)2
+ 11
First make the coefficient of x2 positive
Note there is a change in sign (forgetting to change the sign is a common
mistake), this is because you divided by (-1). When you open the brackets
you should get expression -X2 - 4x + 7
Don’t forget a=1(coefficient of x2) and b=4, this is because
you took the (-1) outside the brackets, c=7
-(b/2a)2 =(4/2)2 = (2)2
- taking the - (2)2 outside the bracket you multiply by the (-1)
outside the bracket. Don’t forget! And that changes to be +(2)2 when
outside the bracket
Again the same idea you have x2 then (2)2, this becomes ( x+2)2
When multiplied by ( -ve)becomes (+ve)
This becomes the answer. If you open the brackets you should
get the quadratic expression (-X2 - 4x + 7)

When “a” is not (1) ( when “a”
is any positive number)
Let’s change the quadratic
expression below to completing
square form.
2X2 - 8x + 11
2(x2- 4x) + 11
2(x2- 4x+ (b/2a)2 –(b/2a)2 ) + 11
2( x2- 4x+ (-2)2 –(-2)2 ) + 11
2( x2- 4x+ (-2)2) + 11–2(-2)2
2( x - 2)2 + 3
= 2( x - 2)2 + 3
First make the coefficient of x2 one (1)
The coefficient of x2 is now one
We add (b/2a) and subtract it, as usual.
From the expression, a=1, b=(- 4) so (-4/2) = (-2)
Never forget multiplying by any number that is outside the bracket,
CONSIDER THE CURVE BELOW
y-axis
( 2,3)
x-axis
2( x - 2)2 + 3
y- co-ordinate of vertex
Opposite of the number you see is the
x- co-ordinate of the vertex, if its (-2) then the x-co-ordinate is 2 and
vice versa.
How to locate the vertex of a
graph a graph using the
complete square form of a
quadratic equation.

How to sketch quadratic graphs
-You need to know whether the curve is facing upwards or downwards
- You need to know the co-ordinates of the vertex.
- Finally, you need to know the x and y- intercepts
Y-intercept
X2 – 5x+ 4=0
X-intercept
(Graph 1) (Graph 2)
Considering the graphs above,
-Is the coefficient of “ x2 ” +ve or -ve, if positive then graph
Is pointing upwards just like graph 1, if negative then pointing
Downwards
-If given equation for graph 1 as x2 – 5x+ 4=0
If you change the quadratic equation in completing
Square form, you get (x-5/2)2-9/4, and therefore the
Co-ordinates of Vertex (for graph 1) is (
5
2
-9
4
,
) and
The same applies for graph 2.

-How to find the co-ordinates of the x-intercept
Recall the equation below,
푥 =
−푏 ± √푏2 − 4푎푐
2푎
The answer will be the x-intercept. Sometimes you may get 2 values of x or one or none. We will look at that in the
following part.
- How to find the co-ordinates of the y-intercept
The y-intercept of the graph is the value of “C “in the quadratic expression ax2 + bx + c

How to determine the number of roots.
푥 =
−푏±√푏2−4푎푐
2푎
푏2 − 4푎푐 Is the discriminant of a quadratic equation
If 푏2 − 4푎푐 > 0 there would be 2 roots (in other words we would have 2 values of x) like graph (a) below
If 푏2 − 4푎푐 = 0, there would be one root (one value of X) like graph ( b) below
If 푏2 − 4푎푐 < 0 there would be no root (no value of x) like graph (c) below
Graph (a) Graph (b) Graph (c)
(>) means greater than
(<) means less than. (We will discuss more about this in inequalities)

Possible questions from this part.
- Show that the graph is lying on the x-axis or intersects only once or a line is a tangent to curve
(tangent intersects only once) for this case you show that the dis criminant is equal to zero.
- Show that the graph the graph is above the x-axis, in this case you could show that the
discriminant is less than zero or you could find the vertex of quadratic equation, the y-value should
be positive.
(We will discuss more about intersection later; don’t worry if I left you behind)
 Inequalities in quadratics
X > 3 means x is greater than 3
X ≥3 means x is greater than or equal to 3
X < 3 means x is less than 3
X ≤ 3 means x is less than or equal to 3
When dividing or multiplying a negative number, direction or inequality changes direction.
Example: if - 2x > 6 but. - 2x > 6
Then x < - 3 (correct) Then x < - 3 ( incorrect)
Note there is change in the inequality sign
And when multiplying an inequity by a negative, this is what happens,

First of all think about it, if I had a simple inequality such as 2 >1, which is very true, 2 is greater
than one, what happens if I multiply by negative both sides, and becomes -2> -1 is the inequality
true? NO! We have to change the direction inequality sign for it to be true and therefore -2 < -1. Hope
you now understand why we change direction of the inequality sign.
 What do you understand from the quadratic inequality x2 – 6x + 8 < 0 ?
You should find the sets of values of x that when plugged in the question, should give you value less
than zero.
Let’s solve it by factorizing method, if you factories you should get
(X-2) (X-4) < 0
Obtain critical values as x=2 and x=4
Draw a simple line as shown below,
Note, the line below is the x- axes, 2 is smaller than 4, so it is written first. (The smaller number should be on the left)
2 4
Critical values
What we are going to do a try and error. We are going to try pick a values less than 2(example 1 , 0 ,-10 etc.) then plug it
in the inequality (X-2) (X-4) < 0 , then pick any value between 2 and 4, try plugging it in and finally pick any
value greater than 4,and do the same.
 If I pick 1 (as a value less than 2) plug it, I end up with 3< 0 ( which is incorrect)
 If I pick 3 (as a value between 2 and 4) I end up with -3< 0 ( which is correct)
 If I pick 5 (as a value greater than 4) I end up with 8< 0 ( which is incorrect)
Therefore the sets of values that obey the inequality are the values of x greater than 2 ( x> 2) and less that 4
(x<4) so answer is 2<x < 4
x2 – 6x + 8 < 0 = (X-2) (X-4) < 0 ,

- Since the inequality was x2 – 6x + 8 < 0 then answer would be (2< x < 4)
- If the inequality was x2 – 6x + 8 ≤ 0, therefore the answer is (2≤ x ≤ 4)
ALTERNATIVELY
-There is an alternative way of doing this, look at the equation given; example x2 – 6x + 8 < 0 it is a quadratic
inequality you know how quadratic graphs looks like don’t you? a parabola? When “a “ is positive it faces upwards,
when “a “is negative it faces downwards.
Now look at the graphs below,
When “a “is positive, When “a “is negative,
Between critical
Values y=0
C D e f
For a graph with “a” as positive, and asked to for a graph with “a” as negative, and asked to
Find the sets of values that would give y-value find the sets of values that would give y-value
Less than zero, obviously it should be the values greater than zero, obviously it should be the values
between the critical values (c< x< d). If asked to between the critical values (e< x< f). If asked to
find sets of value of x that would give “ y ” greater find sets of value of x that would give “ y ” less
that zero the answer is all values greater than “D ” that zero the answer is all values greater than “f ”
and values of x less than “ C ”, answer ( x> D and x<c) and values of x less than “ e ”, answer (x > f and x<e )
Please note that from the equation x2 – 6x + 8 < 0
Is the value of y, that means you are
looking for the set of values that when plugged in the equation, will give you “ y “ being less than zero. In other way we
can say the equation is same as y = x2 – 6x + 8 and therefore y< 0.

Follow the link below for a video lesson of this sub-topic
-If you are given a quadratic equation such as y =2x2 + 6x +7 and a linear equation y = -x + 1.
That means -x + 1= 2x2 + 6x +7, this is equal to 2x2 + 7x +6=0, and from here you can
always solve the it as usual. You can use quadratic formula or factorizing method. Let’s me do
it in both ways just to remind you.
By quadratic formula By Factorizing method
Recall formula:
푥 =
−푏 ± √푏2 − 4푎푐
2푎
A=2, b= 7 and c=6
-Plug in the numbers in the equation above,
푥 = -1.5 or 푥 = -2
Given equation, 2x2 + 7x +6=0
Take a x c= 2 x 6= 12. Then find 2 numbers when
multiplied together gives 12, and when added
give 7(the value of b, in the equation), the
numbers are 4 and 3.
So, , 2x2 + 4x + 3x +6 =0
2X( X+2) + 3(X+2) =0
( 2X+3) + (X+2)= 0
X= - 1.5 and x= -2

-I got two values of x, but remember at first you were given two equations (y =2x2 + 6x +7 and y =
-x + 1).If I plug in the values I got to find the corresponding value of y, in any of the 2 equations, I
should get two values of y, as y= 2.5 and y= 3. In other words it is like saying the two lines intersect at
point (-1.5, 2.5) and (-2, 3). We will do more of this in co-ordinate geometry.

I will use the same example from the syllabus, if give such a question and asked to
solve it don’t be confused by the powers, x4 – 5x2 + 4 = 0 this equation can be written
as (x2)2 – 5(x2) + 4= 0.
-If I let x2 = m, the equation becomes m2 – 5m + 4= 0, does it look familiar now? Of
course it is a quadratic equation. Can you solve a quadratic equation? If no please go
back and learn it one more time.
So , m2 – 5m + 4= 0 , if solved either by factorizing method or quadratic formula, you
should get m =4 or m=1.
But we are asked to solve for the value of x .Remember we let x2 = m, but m=4 or 1
So x2 = 4 or x2 = 1
X= ± 2 or X= ± 1, and this is your answer.

CAMBRIDGE PAST YEAR QUESTIONS
( from CIE)

AS LEVEL QUADRATIC (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

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