On this page we will look at quadratic expressions and how the different factorised forms link to the shape of the graph. We will also get an understanding about how the discriminant affects not only the number of roots of a quadratic equation, but also how we can use it as a tool to help us solve problems with intersections of graphs.
On this page, you should learn to
- Understand the quadratic function \(f(x)=ax^2+bx+c\) and its graph
- Find and use the intercept form \(f(x)=a(x-p)(x-q)\)
- Find and use the vertex form \(f(x)=a(x-h)^2+k\)
- Solve quadratic equations
- Solve quadratic inequalities
- Use the discriminant \(\Delta =b^2-4ac\) to determine the nature of roots
The following videos will help you understand all the concepts from this page
In the following video, we look at quadratic expressions and their different factorised forms, how we might solve quadratic equations and most importantly how this all links to the graphs of the quadratic function.
Notes from the video
In the following video, we will look at a peculiar application of the discriminant.
If a quadratic function f meets the linear function g at a tangent, then the solution to the equation f = g has one (repeated) root. In other words, the f = g gives a quadratic equation and the discriminant = 0
Here is the question that we will consider to look at this case:
The line y = kx + 20 is a tangent to the curve f where f(x) = 12 -2x -2x²
Find the values of k.
Notes from the video
Print from here
Here is a quiz that practises the skills from this page
START QUIZ!
Drag the expressions in the correct place to complete the factorising of the following quadratic expressions
(x - 3) (x + 3) (x + 2) (x - 2) (x - 8)
2x² - 10x + 12 = 2 (x - 3)
3x² - 3x - 16 = 3 (x + 2)
x² - 9 = (x - 3)
3x² + 7x - 6 = (3x - 2)
x² + 6x + 9 =
Find the values for h and k below
x² + 4x + 1 = (x - h)² + k
h =
k =
Careful x² + 4x + 1 = (x + 2)² + 1
Hence h = -2
Find the values of a , h and k
3x² - 12x + 4 = a(x - h)² + k
a =
h =
k =
Match up the function with the correct graph by dragging it in to the correct place
(x + 2)(x - 3) -(x - 2)² -(x + 3)(x - 2) (x + 2)² -(x + 2)(x - 3) -(x +2)² (x + 3)(x - 2) (x - 2)²
| f(x) = |
| f(x) = |
 | f(x) = |  | f(x) = |
| f(x) = |  | f(x) = |
f(x) = a (x - b)(x - c) has x intercepts at x = b and x = c
It is concave up when a>0
It is concave down when a<0
Match up the function with the correct graph by dragging it into the correct place
(x - 2)² + 1 (x - 1)² + 2 (x + 2)² + 1 -(x - 1)² + 2 -(x - 2)² + 1 (x + 1)² - 2 -(x + 2)² + 1 -(x - 1)² - 2
 | f(x) = |  | f(x) = |
| f(x) = |  | f(x) = |
f(x) = a(x - h)² + k has a vertex at the point (h , k)
It is concave up when a>0
It is concave down when a<0
What is the value of the discriminant in the quadratic equation 2x² - 5x + 4 = 0 ?
discriminant =
discriminant = b² - 4ac
= (-5)² - 4x2x4
= 25 - 32
= -7
How many real roots does the function f have?
f(x) = 2x² + 12x + 18
number of roots =
discriminant = b² - 4ac
= (12)² - 4x2x18
= 0
Since discriminant = 0, there is one (repeated root)
We can see this if we write the function in the vertex from
f(x) = 2(x + 3)²
The vertex lies on the x axis at (-3 , 0)
The function below is given by f(x) = ax² + bx + c
Find a , b and c
a =
b =
c =
Since the x intercepts (or zeros) are at x = -3 and x = -1, we know that we can write the function in the form
f(x) = a(x + 3)(x + 1)
Let's expand the brackets
f(x) = a(x² + 4x + 3)
f(x) = ax² + 4ax + 3a
Since the y intercept is at y = 9, then 3a = 9
a = 3
f(x) = 3x² + 12x + 9
The function below is given by f(x) = ax² + bx + c
Find a , b and c
a =
b =
c =
Using the vertex (-3 , 4), we know
f(x) = a (x + 3)² + 4
Use the other point (0 , -14) to find a
-14 = a (0 + 3)² + 4
-14 = 9a + 4
-18 = 9a
a = -2
f(x) = -2 (x + 3)² + 4
f(x) = -2(x² + 6x + 9) + 4
f(x) = -2x² - 12x - 18 + 4
f(x) = -2x² - 12x - 14
The graph of the function f where f(x) = ax² + bx + c is given below
Drag the correct values into the table
negative zero positive
a =
b =
c =
b² - 4ac =
b² - 4ac = 0 because there is one (repeated root).
c is positive. We can see the y intercept is positive.
a is positive. The graph is concave up.
b is positive. We can write the function in the vertex form f(x) = a(x - h)² . Since the vertex crosses the x axis where x<0, h must be negative. If you expand these brackets you will see that the x term is positive.
The graph below is the function f(x)=a(x-h)²+k
The graph pass through the point (0,-9) and has vertex (2,3)
a) Write down the value of a, h and k
b) Find f(x) giving your answer in the form f(x) = Ax²+Bx+C
Hint
Full Solution
Let f(x) = 2x²+12x +11
The function can also be expressed in the form f(x) = a(x-h)² +k
a) Find the equation of the axis of symmetry
b) Write down the value of h
c) Write down the value of k
Hint
Full Solution
Let f(x) = x² + 2px + (3p+4)
Find the value of p so that f(x)=0 has two equal roots.
Hint
Full Solution
Let f(x)=2x² + kx + 1 and g(x) = -x – 1.
The graphs of f and g intersect at two distinct points.
Find the possible values of k
Hint
Full Solution
Just for Fun
If you understand the form y=a(x - p)(x - q) of a quadratic equation, then you should be able to play and win this game of angry birds. The angry bird starts at (0 , 0), the pig is at (10 , 0) and the bird must pass through (5,4).
Enter the equation in the space below and press play.
How much of Quadratics have you understood?
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