 # Solve Quadratic Equations by Factorisation

In this worksheet, students will learn how to solve a quadratic equation of the form x² + bx + c = 0 by factorising. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:   ### QUESTION 1 of 10

In this activity, we will be looking at quadratic equations, which cannot be solved by the 'balancing' method. For example:

x² - 3x - 18 = 0

For these quadratic equations, there are several methods which can be used, but in this activity, we will use the method of factorisation. How to factorise a quadratic expression is covered in another activity, and here we assume you have completed that activity, but if you haven't it would be best if you do that before continuing here.

So, the equation x² - 3x - 18 = 0 can be factorised into two brackets as (x - 6)(x + 3) = 0. Remember, the numbers in the brackets must multiply to make -18 and add to make -3. Now, remember that each set of brackets stand for a number and these are multiplied together to give the answer zero. So, if I tell you that I have multiplied together two numbers and the answer is zero can you tell me anything about my two numbers? Well, you should be able to tell me that at least one of my numbers is zero itself. Going back to my equation, that means that either (x - 6) = 0 or (x + 3) = 0 (they cannot both be zero). But if x - 6 = 0 then x must be 6 so x = 6 is a solution. Alternatively, if x + 3 = 0 then x = -3 and this is another solution. So there are two possible solutions to this quadratic equation, x = 6 and x = -3. it is a feature of quadratic equations that they nearly always have two solutions.

Here's another example, this time I'll just include the lines of working you would need to show.

n² - 10n + 25 = 0

(n - 5)(n - 5) = 0

Either n - 5 = 0 or n - 5 = 0

So n = 5 or n = 5

Now in this case, because both sets of brackets are the same both solutions are the same so really there is just one solution n = 5. But this is unusual, usually there will be two different sets of brackets with two different solutions.

Here's one more example, again slightly different.

2y² - 4y = 0

This one will not factorise into two brackets, but we can take out a common factor of 2y from each term on the left.

2y(y - 2) = 0

Either 2y = 0 or y - 2 = 0

So y = 0 or y = 2

Ok, now you should be ready to try the questions.

Below you will see the steps to solving the quadratic equation

x² - 7x + 12 = 0

Match the steps to the correct order.

## Column B

1st
Either x - 3 = 0 or x - 4 = 0
2nd
So x = 3 or x = 4
3rd
x² - 7x + 12 = 0
4th
(x - 3)(x -4) = 0

Below you see the steps for solving the equation

x² + 9x + 20 = 0

Fill in the spaces in each line to make the solution complete.

## Column B

1st
Either x - 3 = 0 or x - 4 = 0
2nd
So x = 3 or x = 4
3rd
x² - 7x + 12 = 0
4th
(x - 3)(x -4) = 0

Solve the equation

m² -5m - 6 = 0

Choose the correct pair of solutions from those given below.

m = 3 or m = -2

m = -3 or m = 2

m = 6 or m = -1

m = -6 or m = 1

From the choices given below, select the two correct solutions to the equation

y² + 9y + 18 = 0.

y = -9

y = -2

y = 3

y = -3

y = 6

y = -6

Which of the solutions below are correct for the equation
x² - 7x = 0?

x = 0

x = 3

x = 7

x = 4

x = -7

Fill in the spaces in the working below for solving the equation

p² - 6p + 8 = 0

x = 0

x = 3

x = 7

x = 4

x = -7

The following quadratic equation only has one solution.

a² - 14a + 49 = 0

Choose the solution from the following options.

a = -7

a = 7

a = 14

a = -14

The solution x = 2 is one of the correct solutions to which of the following equations? You can choose more than one.

x² - 4x + 4 = 0

x² + 4x + 4 = 0

x² + 9x - 22 = 0

x² - 21x + 38

x² - 2x + 2 = 0

Solve the equation

m² + 7m -30 = 0

Fill your solutions in the spaces below.

x² - 4x + 4 = 0

x² + 4x + 4 = 0

x² + 9x - 22 = 0

x² - 21x + 38

x² - 2x + 2 = 0

Can you solve the following equation?

x² + 6x - 91 = 0

Fill in the spaces below with your solution.

x² - 4x + 4 = 0

x² + 4x + 4 = 0

x² + 9x - 22 = 0

x² - 21x + 38

x² - 2x + 2 = 0

• Question 1

Below you will see the steps to solving the quadratic equation

x² - 7x + 12 = 0

Match the steps to the correct order.

## Column B

1st
x² - 7x + 12 = 0
2nd
(x - 3)(x -4) = 0
3rd
Either x - 3 = 0 or x - 4 = 0
4th
So x = 3 or x = 4
EDDIE SAYS
To solve the equation x² - 7x + 12 = 0 first we factorise: (x - 3)(x -4) = 0 then equate each bracket to zero: Either x - 3 = 0 or x - 4 = 0 finally write down the values of x: So x = 3 or x = 4
• Question 2

Below you see the steps for solving the equation

x² + 9x + 20 = 0

Fill in the spaces in each line to make the solution complete.

EDDIE SAYS
These are the steps for solving the equation: x² + 9x + 20 = 0 (x + 5)( x + 4) = 0 Either x + 5 = 0 or x + 4 = 0 So x = -5 or x = -4 Did you get them all correct?
• Question 3

Solve the equation

m² -5m - 6 = 0

Choose the correct pair of solutions from those given below.

m = 6 or m = -1
EDDIE SAYS
Here are the steps for solving this equation. m² -5m - 6 = 0 (m - 6)(m + 1) = 0 Either m - 6 = 0 or m + 1 = 0 So m = 6 or m = -1 It is easy to make the mistake of using -3 and 2 or 3 and -2 here because they multiply to make -6 but when you add them they come to -1 or +1 respectively.
• Question 4

From the choices given below, select the two correct solutions to the equation

y² + 9y + 18 = 0.

y = -3
y = -6
EDDIE SAYS
Here are the steps for solving this equation y² + 9y + 18 = 0 (y + 3)(y + 6) = 0 Either y + 3 = 0 or y + 6 = 0 So y = -3 or y = -6 How are you doing? Getting the hang of them?
• Question 5

Which of the solutions below are correct for the equation
x² - 7x = 0?

x = 0
x = 7
EDDIE SAYS
Here are the steps for solving this equation x² - 7x = 0 x(x - 7) = 0 Either x = 0 or x - 7 = 0 So x = 0 or x = 7 How did you do with this one? Did you spot that it factorises into a single bracket?
• Question 6

Fill in the spaces in the working below for solving the equation

p² - 6p + 8 = 0

EDDIE SAYS
Here is the full working for this one. p² - 8p - 20 = 0 (p - 10)(p + 2) Either p - 10 = 0 or p + 2 = 0 So p = 10 or p = -2
• Question 7

The following quadratic equation only has one solution.

a² - 14a + 49 = 0

Choose the solution from the following options.

a = 7
EDDIE SAYS
Here is the full solution to this one. a² - 14a + 49 = 0 (a - 7)(a - 7) = 0 Either a - 7 = 0 or a - 7 = 0 So a = 7
• Question 8

The solution x = 2 is one of the correct solutions to which of the following equations? You can choose more than one.

x² - 4x + 4 = 0
x² + 9x - 22 = 0
x² - 21x + 38
EDDIE SAYS
We can check each equation by substituting x = 2 into them to see if the answer is zero. x² - 4x + 4 = 0 → 2² - 4x2 + 4 = 4 - 8 + 4 = 0 (correct) x² + 4x + 4 = 0 → 2² + 4x2 + 4 = 4 + 8 + 4 = 16 (wrong) x² + 9x - 22 = 0 → 2² + 9x2 - 22 = 4 + 18 - 22 = 0 (correct) x² - 21x + 38 → 2² - 21x2 + 38 = 4 - 42 + 38 = 0 (correct) x² - 2x + 2 = 0 → 2² - 2x2 + 2 = 4 - 4 + 2 = 2 (wrong) So there are 3 correct equations. DId you get them all?
• Question 9

Solve the equation

m² + 7m -30 = 0

Fill your solutions in the spaces below.

EDDIE SAYS
Here's the full solution to this equation m² + 7m -30 = 0 (m + 10)(m -3) = 0 Either m + 10 = 0 or m - 3 = 0 So m = -10 or m = 3 How's it going? Just one more left now.
• Question 10

Can you solve the following equation?

x² + 6x - 91 = 0

Fill in the spaces below with your solution.

EDDIE SAYS
Last one now. This has some trickier numbers but it helps to know that 91 = 13 x 7. Here's the solution. x² + 6x - 91 = 0 (x - 7)(x + 13) = 0 Either x - 7 = 0 or x + 13 = 0 So x = 7 or -13 That's it. You're done now.
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