Quadratic Equations
Quadratic Equations
An Introduction
THE STEM CELL
Friday, February 28, 2025
This slide carries a
very important
message
Before we start
(c) Glasbergen.com
(Reproduced with
permission)
THE STEM CELL
THE STEM CELL
Friday, February 28, 2025
Quadratic Equations
An Introduction
THE STEM CELL
Friday, February 28, 2025
We have already seen linear equations (equations with single-power values for the variable 'x') and how these
will produce graphs which are a straight line, the general formula is:
Now we will take a look at the next ‘order’ of algebraic expressions and equations, those which hold an instance
of the variable ‘x’ to a second power:
Where ‘a’, ‘b’ and ‘c’ are numbers with integer values.
These equations have usually two solutions, that is, two numbers whose values satisfy the
equation. These numbers can be the same value, or different.
There is another type of solution involving ‘complex’ numbers but we will not look at these here.
THE STEM CELL
Friday, February 28, 2025
So, how do we solve such equations?
There are several ways, but first of all we will look at solution by factorisation where we will ‘split’ the
expression into two bracketed expressions for ‘x’ and obtain values for ‘x’ such that one of the two pairs of
brackets becomes equal to ZERO. This will provide one of the solutions as the expression in whole becomes
equal to zero.
We need to obtain two numbers whose SUM comes to the value of the constant ‘b’ shown
in green, but whose product is the same as the product ‘ac’
Let’s look at an example:
THE STEM CELL
Friday, February 28, 2025
Let’s look at an example:
Using the general equation we can see that we are looking for two numbers whose SUM is +1 but whose
product is -12 …… pause for a moment to make sure you can see why this is the case.
Step 1 – write out the factors of 12, don’t worry about signs at this stage.
Step 2 – inserting + and – can you locate two numbers whose SUM would be +1 AND whose PRODUCT would
be -12 ?
-3 and +4
Step 3 – we can now split the linear expression (the one involving ‘b’) into two:
THE STEM CELL
Friday, February 28, 2025
Step 4 – pair off the expressions and factorise:
Step 5 – Tidy up:
To actually find the values of ‘x’ that will satisfy the original equation we now look for individual values that
will render each bracket equal to 0:
THE STEM CELL
Friday, February 28, 2025
The values that satisfy the equation are called the ‘roots’
of the Quadratic.
In this case, two real numbers (this isn’t always the case)
Now have a go at these examples
THE STEM CELL
Friday, February 28, 2025
a=+1, b=+6 and c=+8
Factors of 6 are 1,2,4
Selection to satisfy ‘ac’ and ‘b’ requirements are +2 and +4
Pair off:
Factorise:
Rearrange:
THE STEM CELL
Friday, February 28, 2025
a=+1, b=-4 and c=-21
Factors of 21 are 1,3,7
Selection to satisfy ‘ac’ and ‘b’ requirements are +3 and -7
Pair off:
Factorise:
Rearrange:
THE STEM CELL
Friday, February 28, 2025
a=+2, b=-5 and c=-12
Factors of 24 are
1,2,3,4,6,8,12
Selection to satisfy ‘ac’ and ‘b’ requirements are +3 and -8
Pair off:
Factorise:
Rearrange:
Quadratic equations involve variables raised to the power of 2, requiring a different approach from linear equations. This introduction explains the nature of quadratic equations and how to solve them through factorization. Through examples and step-by-step instructions, readers can learn how to identify the solutions of quadratic equations for both real and complex numbers.
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Presentation Transcript
Friday, February 28, 2025 Quadratic Equations An Introduction THE STEM CELL
Before we start This slide carries a very important message (c) Glasbergen.com (Reproduced with permission) THE STEM CELL
Friday, February 28, 2025 Quadratic Equations An Introduction THE STEM CELL
Friday, February 28, 2025 We have already seen linear equations (equations with single-power values for the variable 'x') and how these will produce graphs which are a straight line, the general formula is: Now we will take a look at the next order of algebraic expressions and equations, those which hold an instance of the variable x to a second power: Where a , b and c are numbers with integer values. These equations have usually two solutions, that is, two numbers whose values satisfy the equation. These numbers can be the same value, or different. There is another type of solution involving complex numbers but we will not look at these here. THE STEM CELL
Friday, February 28, 2025 So, how do we solve such equations? There are several ways, but first of all we will look at solution by factorisation where we will split the expression into two bracketed expressions for x and obtain values for x such that one of the two pairs of brackets becomes equal to ZERO. This will provide one of the solutions as the expression in whole becomes equal to zero. We need to obtain two numbers whose SUM comes to the value of the constant b shown in green, but whose product is the same as the product ac Let s look at an example: THE STEM CELL
Friday, February 28, 2025 Let s look at an example: Using the general equation we can see that we are looking for two numbers whose SUM is +1 but whose product is -12 pause for a moment to make sure you can see why this is the case. Step 1 write out the factors of 12, don t worry about signs at this stage. 1 2 3 4 6 12 Step 2 inserting + and can you locate two numbers whose SUM would be +1 AND whose PRODUCT would be -12 ? -3 and +4 Step 3 we can now split the linear expression (the one involving b ) into two: THE STEM CELL
Friday, February 28, 2025 Step 4 pair off the expressions and factorise: Step 5 Tidy up: To actually find the values of x that will satisfy the original equation we now look for individual values that will render each bracket equal to 0: THE STEM CELL
Friday, February 28, 2025 The values that satisfy the equation are called the roots of the Quadratic. In this case, two real numbers (this isn t always the case) Now have a go at these examples THE STEM CELL
Friday, February 28, 2025 a=+1, b=+6 and c=+8 Selection to satisfy ac and b requirements are +2 and +4 Factors of 6 are 1,2,4 Pair off: Factorise: Rearrange: THE STEM CELL
Friday, February 28, 2025 a=+1, b=-4 and c=-21 Factors of 21 are 1,3,7 Selection to satisfy ac and b requirements are +3 and -7 Pair off: Factorise: Rearrange: THE STEM CELL
Friday, February 28, 2025 Factors of 24 are 1,2,3,4,6,8,12 a=+2, b=-5 and c=-12 Selection to satisfy ac and b requirements are +3 and -8 Pair off: Factorise: Rearrange: THE STEM CELL
