Section 2: Quadratic Functions
Introduction
This section will cover how to: Correctly evaluate quadratic functions given values of x
 Produce a table of coordinates from y = ax² + bx + c
 Plot the coordinates on a graph and use the graph to identify roots of the quadratic
 Expand pairs of brackets (px+q)(rx+s) to give quadratic expressions
 Factorise quadratic expressions into two brackets
 Use "difference of two squares" a²  b² = (a+b)(a–b) and factorising ax² + bx = x(ax+b)
 Solve quadratic equations by factorising
 Solve quadratic equations using the formula
Quadratic Graphs
Definitionsquadratic expression  an expression of the form ax² + bx + c where a, b and c are numbers and x is the variable (note that b and c can be zero, but a must be nonzero so that there is always an x² term) e.g. x² + 5x + 6, 3x² – 3x + 17, 8x², 4x² – 25, 12x² – 30x  
quadratic equation  an equation which can be written with a quadratic expression on one side and zero on the other e.g. x² + 5x + 6 = 0, 3x² – 3x + 17 = 0, 4x² – 25 = 0  
note that quadratic equations sometimes need rearranging to get zero on one side e.g. x² = 3x – 2 can be rearranged to give x² – 3x + 2 = 0  
quadratic function  a name normally given to a function of the form y = ax² + bx + c which is used to calculate y coordinates from given x coordinates so that these can be plotted on a graph  
roots of a quadratic  the roots of a quadratic equation are the solutions to that equation e.g. the roots of the equation x² + 5x + 6 = 0 are x = –2 and x = –3  
the roots of a quadratic expression are the values which make it equal zero e.g. the roots of the expression x² + 5x + 6 are x = –2 and x = –3  
on a graph, the roots correspond to where the curve crosses the xaxis e.g. the curve of y = x² + 5x + 6 crosses the xaxis at (–2,0) and (–3,0) 
To plot the graph of y = x² – 2x – 3 for values of x between –2 and 4, we start by calculating a table of values: 

For example for the x coordinate of –2 we calculate the y coordinate as follows:

You need to be especially careful when calculating the x² term. The two main things to watch out for are:

Now that we have a table of values, we can plot each ( x , y ) pair as coordinates on a set of axes, as shown below: 

Finally we can use the graph to identify the roots of the quadratic expression x² – 2x – 3. They are x = –1 and x = 3. These are the solutions of the quadratic equation x² – 2x – 3 = 0 and are the points where the curve crosses the xaxis. 

Work out the answer to each part and then click on the button marked
to see if you are correct.
For the graph, you can plot the curve yourself on paper and then check your curve against the one on the screen. 
Plot the quadratic curve y = ½x² – 4x + 6 for x = 0 to 6 and use it to identify the solutions of ½x² – 4x + 6 = 0 (a) Start by completing a table of values for x = 0 up to x = 6 (b) Now plot the points on an appropriate set of axes and join them with a smooth curve (c) Finally, use your curve to identify the solutions of ½x² – 4x + 6 = 0

Expanding and Factorising Quadratics
Expanding Pairs of BracketsIf you expand brackets of the form (px + q)(rx + s) you will get a quadratic expression.  
To expand brackets in this form, simply multiply each term in the first bracket by every term in the second bracket, then add up the resulting terms and simplify if necessary.  
e.g.
= x² + 8x + 15 
e.g.
= x² – 2x – 24 
e.g.
= 2x² + x – 21 
e.g.
= 20x² – 23x + 6 
Factorising a quadratic is the reverse of expanding a pair of brackets; not every quadratic can be factorised, but the majority of those which can will end up as two brackets multiplied together. In every case we can easily check if we have factorised correctly by multiplying out the brackets again.  
(a) Simple Factorisation
 
(b) Harder Factorisation
 
(c) Special Cases

Work out the answer to each question then click on the button marked
to see if you are correct.

Expand the brackets and simplify: (a) ( x + 5 )( x – 7 )
= x² – 2x – 35 (b) ( 3x – 5 )( 4x – 1 )
= 12x² – 23x + 5  
Factorise completely: (a) x² + 10x + 9
( x + 1 )( x + 9 )
(b) x² – 7x – 30
( x + 3 )( x – 10 )
(c) 2x² + 7x + 5
( 2x + 5 )( x + 1 )
(d) 20x² – 117x – 50
( 4x – 25 )( 5x + 2 )
(e) 9x² – 1
( 3x + 1 )( 3x – 1 )
(f) 20x – 12x²
4x( 5 – 3x )

Solving Quadratic Equations by Factorisation
Basic Principles
Consider the equation ab = 0. The only way this can be true is if a = 0 or b = 0. 
Now consider the quadratic equation ( x + 2 )( x – 3 ) = 0. In a similar way to the equation above, the only way this equation can be true is if x + 2 = 0 or if x – 3 = 0. From x + 2 = 0 we get x = –2 and from x – 3 = 0 we get x = 3. The two solutions of the equation ( x + 2 )( x – 3 ) = 0 are x = –2 and x = 3 (try them). 
Finally consider the quadratic equation x² – 8x + 15 = 0. By factorising we can rewrite this equation as ( x – 3 )( x – 5 ) = 0. For this to be true, either x – 3 = 0 or x – 5 = 0. This gives two solutions for the original equation: x = 3 or x = 5. We can check these by substituting back in: (3)² – 8×(3) + 15 = 0 and (5)² – 8×(5) + 15 = 0 
We can use the principles above to solve any quadratic equations where the quadratic will factorise. In each case we factorise the quadratic, then set each bracket in turn equal to zero to find the solutions. 

Make sure you understand all the following examples before moving on:  
e.g. x² + 4x – 21 = 0 ( x + 7 )( x – 3 ) = 0 x + 7 = 0 or x – 3 = 0 x = – 7 or x = 3 
e.g. x² – 49 = 0 ( x + 7 )( x – 7 ) = 0 x + 7 = 0 or x – 7 = 0 x = – 7 or x = 7 
e.g. 4x² – 15x + 9 = 0 ( 4x – 3 )( x – 3 ) = 0 4x – 3 = 0 or x – 3 = 0 x = ¾ or x = 3 
e.g. 4x² – 4x + 1 = 0 ( 2x – 1 )( 2x – 1 ) = 0 2x – 1 = 0 or 2x – 1 = 0 x = ½ (only one solution  also called equal roots) 
e.g. x² – 6x + 10 = 0 (the quadratic does not factorise so can't be solved by factorisation) 
e.g. 8x² = 2x 8x² – 2x = 0 2x( 4x – 1 ) = 0 2x = 0 or 4x – 1 = 0 x = 0 or x = ¼ 
Work out the answer to each question then click on the button marked
to see if you are correct.

Solve by factorising: (a) x² – x – 20 = 0
Start by factorising:
( x + 4 )( x – 5 ) = 0 x + 4 = 0 or x – 5 = 0 x = – 4 or x = 5 (b) 16x² – 9 = 0
Use the difference of two squares:
(4x)² – (3)² = 0 ( 4x + 3 )( 4x – 3 ) = 0 4x + 3 = 0 or 4x – 3 = 0 x = – ¾ or x = ¾ (c) 4x² – 24x + 36 = 0
You can factorise this but all three terms have a common factor of 4 so it's easiest to divide through by 4 first:
x² – 6x + 9 = 0 ( x – 3 )( x – 3 ) = 0 x – 3 = 0 or x – 3 = 0 x = 3 
Solving Quadratic Equations using the Formula
The Quadratic Formula
 
If a quadratic factorises then it is always quicker and easier to solve a quadratic equation that way, especially without a calculator. Nonetheless, the quadratic formula provides an alternative method for solving quadratic equations. 
The stages you should go through when using the formula are as follows:
 
Look at this example: x² + 4x – 21 = 0  

e.g. 9x² – 9x + 2  
Note that this question would have been much easier using factorisation!  
e.g. 2x² – 12x + 18  
Note that there will only ever be one solution when b² – 4ac = 0.  
e.g. x² – 6x + 10  
Note that there will never be any solutions when b² – 4ac is negative. 
Work out the answer to each question then click on the button marked
to see if you are correct.

Solve these quadratic equations using the formula: (a) x² + 8x + 7 = 0
(b) 3x² – 4x + 2 = 0
(c) 20x² – 9x + 1 = 0

Exercise
Question 1
(a) Fill in the table of values below for the quadratic formula:
y = 2x² – 4x – 6
Substitute each x value in turn into the formula to get the y value. Remember that if you square a negative number you get a positive result. See the boxes above for the correct answers.
(b) Plot the graph of y = 2x² – 4x – 6 on the grid on the righthand side. To plot the graph, click the button, then click each point you want to plot on the graph. When you have finished, click the button and a smooth line will be drawn in. You can then check if your answer is correct using the buttons below. If you need to have another go, click again. Each pair of values for x and y should be plotted as coordinates ( x , y ). The correct graph has been plotted for you on the grid.
(c) Use your graph to identify the two roots of the quadratic equation:
2x² – 4x – 6 = 0. The roots of the equation are the xvalues which make the quadratic evaluate to zero. The quadratic evaluates to zero where it crosses the xaxis at (–1,0) and (3,0) so the solutions are x = –1 and x = 3. 
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Question 2
Multiply out the brackets below, simplifying your answers:

Question 3
Factorise the following expressions completely:

Question 4
Solve the quadratic equations below by factorising, simplifying your answers where necessary.If you think there is only one solution, place a dash (–) in the second solution box.
If you think there are no solutions then put a dash in both boxes.

Question 5
Solve the quadratic equations below using the formula, simplifying your answers where necessary.If you think there is only one solution, place a dash (–) in the second solution box.
If you think there are no solutions then put a dash in both boxes.

Question 6
Solve the quadratic equations by any method you wish, simplifying your answers where necessary.If you think there is only one solution, place a dash (–) in the second solution box.
If you think there are no solutions then put a dash in both boxes.
