1
Quadratics – What are they?
2
x
+ 3x – 4
2
y
– 4y + 2
1–

Quadratics are
equations where the
highest power is 2.

What shape do their
graphs make?
2
x
Maths revision course by Miriam Hanks
2
Quadratic graphs
Quadratics make a shape
called a parabola
It is a smiley face if the
coefficient of x2 is
positive, or a sad face if
it’s negative.
Maths revision course by Miriam Hanks
3
Quadratics – Sketching graphs
Any of these will help:
 Identify shape:
or

Find where it crosses the
y-axis by putting x = 0
 Find where it crosses the
x-axis by putting y = 0
 Complete the square to
find the turning point
 Differentiate to find the
turning point
Maths revision course by Miriam Hanks
4
Completing the square


If there is a coefficient of x2, take it out as a factor
of the first 2 terms.
eg
2x2 + 12x – 5
= 2[x2 + 6] – 5
Now insert a new bracket, move the
“squared” sign to the outside of it, halve the
number inside, and square and subtract it:
= 2[(x + 3)2 - 9] - 5
Maths revision course by Miriam Hanks
5
Completing the square

Next remove the outer bracket,
remembering to multiply by your factor
= 2(x + 3)2 - 18 – 5

Finally, tidy up the last two terms:
= 2(x + 3)2 - 23
Maths revision course by Miriam Hanks
6
Completing the square
y =2(x + 3)2 - 23

How does this help draw the
graph?

The turning point is
(-3, -23)
Maths revision course by Miriam Hanks
7
Solving a quadratic by using the formula.
The quadratic formula is not given to you in the Higher
maths exam, so you should learn it:
 b  b  4ac
x
2a
2
Maths revision course by Miriam Hanks
8
The discriminant =
2
b
– 4ac
If b2 – 4ac = 0, then there is one real root (or two
equal roots)
Turning point is on x-axis
If b2 – 4ac > 0, then there are 2 real roots
Graph crosses x-axis twice
If b2 – 4ac < 0, then there are no real roots
Graph does not cross the x-axis
Maths revision course by Miriam Hanks
9
Quadratic inequalities


If your quadratic has a > or < sign,
Start solving it as normal, but when you get
the 2 solutions, decide on the direction of
the < or > arrows by drawing a diagram.
If you don’t draw the diagram, you will not
get full marks in the exam.
Maths revision course by Miriam Hanks
10
Quadratic inequalities example 1

Eg Solve x2 + 3x – 4 < 0 gives
(x + 4) (x – 1) and so we mark x = -4 and
x = 1 on the diagram:
Since the original equation
had a “< 0”, we look at
where the graph is below
-4
1
the x-axis. Final answer:
-4 < x < 1
Maths revision course by Miriam Hanks
11
Quadratic inequalities example 2

Eg Solve x2 + 3x – 4 > 0 gives
(x + 4) (x – 1) and so we mark x = -4 and x
= 1 on the diagram:
Since the original equation
had a “> 0”, we look at
where the graph is above
-4
1
the x-axis. Final answer:
x < -4 and x > 1
Maths revision course by Miriam Hanks
12
Quadratics in real life
Quadratics are used to
make satellite dishes,
suspension bridges and
torches, all of which have
a parabolic curve.
Maths revision course by Miriam Hanks
13
Quadratics in real life
When you throw an object, the path it takes is
also a parabola:
Click for video clip
Maths revision course by Miriam Hanks