Quadratic Equations
It has been my experience that students overcomplicate the topic of quadratic equations.
There are a number of different ways to solve quadratic equations in addition to factorising, such as: the ‘difference of two squares’ (Level 6); the quadratic formula (Level 7), plotting the graphs of quadratic functions (Levels 5 – 7) and completing the square (Level 8). However, in this blog I am focusing purely on factorising.
My advice is to get familiar with what a quadratic expression looks like and the steps I have shown below. There are things you need to be familiar with first and they are covered in the next section. Most of the errors I come across are where the student has either missed or got muddled with the rules concerning negative values or made a mistake rearranging the equation.
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What you need to know first when tackling QUADRATIC EQUATIONS:
- Order of operations for solving mathematical problems BIDMAS (including inverse relationships):
- Brackets
- Indices
- Divide and /or Multiply
- Add and / or Subtract
- How to apply the four operations in fractions (i.e. addition, subtraction, multiplication and division)
- How to rearrange and solve equations
- How to expand brackets and simplify algebraic expressions
- Express problems / situations using algebraic notation
- How to factorise simple expressions (‘putting the brackets in’)
Understanding Simultaneous Equations
What do we mean by a quadratic expression?
A quadratic expression is one where the highest power of x in the expressions is x squared (x2). Quadratic equations can be written in the form of:
ax2 + bx + c = 0
where a, b and c are real numbers and a is not equal to zero. This means that the values of a, b and c can be positive as well as negative. They can also be decimals or fractions.
The solutions to a quadratic equation are known as roots and there are generally two to find.
Note: The roots are also the points on a graph where the quadratic curve cuts the x-axis.
How to factorise quadratic equations
Step 1: Rearrange (if necessary)
Make sure the quadratic equation in the form ax2 + bx + c = 0 when factorising.
This may mean you need to rearrange the equation you are given before you start.
Example: X2 +2X = 15 rearrange to X2 + 2X – 15 = 0
Step 2: Prepare Brackets
Note the values of a, b and c. and write down the initial brackets (as factorising is putting the brackets in).
Example: In the expression X2 + 2X – 15 = 0 a = 1, b = 2, c = -15. As a is = 1, the brackets will be:
(X )(X ) = 0
Step 3: Find the values
When a is equal to 1 (as in this example), we can use the following method:
X2 + ( j+k )X + ( jxk )=0
where j and k are two real numbers.
The middle term (“b”) is the sum of j and k.
The third term (“c”) is the product of j multiplied by k
Example: X2 + 2X – 15 = 0 where a = 1, b = 2 and c = -15
j + k = 2
j x k = -15
so…….. 5 x -3 = -15 and 5 + -3 = 2
Therefore the factorised quadratic expression is ( X + 5 )( X – 3) = 0
Step 4: Check
It is important to check your answers are correct by expanding the brackets and simplifying the equation to make sure you get the original expression.
Example: (X + 5 )( X – 3 ) = X2 + 5X – 3X – 15
= X2 + 2X – 15
Step 5: Solve
By setting each bracket to zero
Example:
( X + 5 ) = 0 so X = -5
( X – 3 ) = 0 so X = 3
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What to watch out for:
- This is likely to be the method used in a non-calculator paper
- When c = 0, the quadratic will look like this: X2 + bX = 0
Example: X2 – 10X = 0 which will factorise to X(X-10) = 0
Solutions are X=0 and X=10
- When b=0 !!!!!
- Take care where the coefficient of X2 is greater than 1 (i.e. 3X2) where a is bigger than 1
Example: Factorise 3X2 + 32X = – 20
Step 1: Rearrange to 3X2 + 32X + 20 = 0
Step 2: Prepare brackets. a = 3, b = 32, c = 20 so……….(3X )(X ) = 0
Step 3: Find the values:
j x k = 20 and j + 3k = 32 so………2 x 10 = 20 and 2 + 3×10 = 32
( 3X + 2 )( X + 10 ) = 0
Step 4: Check
( 3X + 2 )( X + 10 ) = 3X2 + 30X + 2X + 20
= 3X2 + 32X + 20
Step 4: Solve
( 3X + 2 ) = 0 , X = -2/3
( X + 10 ) = 0, X = -10
What are the examiners looking for?
- Evidence that a correct method has been used – so show your working out!!!
- The correct values in the brackets
- The final solutions for the two values of X (so don’t forget to state them clearly at the end).
A Bit About Karen
A Bit About Karen Karen is an experienced, PGCE qualified teacher of Mathematics at Key Stage 3 and above, including the International GCSE. She has been an Examiner and Moderator for Edexcel and therefore has good insight into what examiners are looking for and with exam techniques.
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Karen is a skilled creative thinker and problem solver with a passion for identifying untapped talent in individuals so they are able to make a lasting transformational change into the best version of themselves. She provides a safe environment that is non-judgemental, encouraging, positive and challenging, with the freedom to experiment.
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