Solving Quadratics Through Factorising Worksheets, Questions and Revision

Solving Quadratics Through Factorising Worksheets, Questions and Revision

GCSE 4 - 5AQAEdexcelOCRWJECFoundationAQA 2022Edexcel 2022OCR 2022WJEC 2022

Solving Quadratic Equations by Factorising

Quadratics are algebraic expressions that include the term, x^2, in the general form,

ax^2 + bx + c

If you are on the foundation course, any quadratic equation you’re expected to solve will always have a=1, with all terms on one side and a zero on the other. If you are on the higher course, you may have to do some rearranging in order to get all the terms on one side. Make sure you are happy with the following topics before continuing.

Level 4-5 GCSE AQA Edexcel OCR WJEC

Solving through Factorising (a=1)

We can solve quadratics through factorising by following these 4 easy steps.

Example: Solve the quadratic equation, \textcolor{blue}{ x^2-3x=-2} by factorisation.

Since the right-hand side of the equation is zero, the result of multiplying the two brackets together on the left-hand side must be zero. Therefore, at least one of the brackets must be equal to zero. So, in summary,

If (x-2)(x-1)=0, then either (x-2)=0 or (x-1)=0.

Equation 1:

\begin{aligned}(+2) \,\,\,\,\,\,\,\,\, x - 2 &= 0 \\  x &= 2\end{aligned}

Equation 2:

\begin{aligned}(+1) \,\,\,\,\,\,\,\,\, x - 1 &= 0 \\  x &= 1\end{aligned}

The final roots are:

x = 2 \,\,\,\,\text{and}\,\,\,\, x=1

Note: The solutions to a quadratic equation are also called roots, because they correspond to where a quadratic graph crosses the x-axis.

Level 4-5 GCSE AQA Edexcel OCR WJEC
Level 6-7 GCSE AQA Edexcel OCR WJEC

Solving through Factorising (a>1)

Solve the following quadratic equation through factorising 2x^2-3x-9=0

Step 1: Rearrange the given quadratic so that is it equal to zero (=0)

This quadratic is already equal to 0 so there is nothing more to do.

Step 2: Factorise the quadratic,

(2x+3)(x-3)=0

Step 3: Form two linear equations

2x+3=0 \,\,\text{and}\,\, x-3=0

Step 4: Solve the equations to find the roots of the equation

Equation 1:

\begin{aligned}(-3)\,\,\,\,\,\,\,2x+3&=0 \\ (\div2)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x&=-3 \\ \,\,\,\,\,\,\,\,\, x& = -\dfrac{3}{2}\end{aligned}

Equation 2:

\begin{aligned}(+3)\,\,\,\,\,\,\,\,\,x-3&=0 \\ x &=3 \end{aligned}

So, the 2 solutions to the equation are x=3 and x=-\frac{3}{2}.

Level 6-7 GCSE AQA Edexcel OCR WJEC
Level 6-7 GCSE AQA Edexcel OCR WJEC

Quadratic Equations and Sketching Graphs

It is possible to use factorisation to allow you to sketch a quadratic graph.

Example: Use factorisation to find the roots of x^2 -2x -3=0 and hence sketch the quadratic.

Like always, the equation needs to be =0

Quadratic Equation Graph Sketch Using Factorisation Example
Quadratic Equation Graph Sketch Using Factorisation Example

Step 1: First we need to factorise the left hand side of the equation

(x-3)(x+1) = 0

Step 2: Solve the quadratic as show in the above examples

\begin{aligned}(+3)\,\,\,\,\,\,\,\,\,x-3&=0 \\ x&=3\end{aligned}

\begin{aligned}(-1) \,\,\,\,\,\,\,\,\,x+1&=0 \\ \,\,\,\,\,\,\,\,\, x&=-1\end{aligned}

Step 3: Find the coordinates of the roots. We know this equation has the solutions x= 3 and x=-1

When we set the equation to 0, y=0. This means we can form the two coordinates

(3,0) and (-1,0)

Step 4: Identify the y-intercept. This is the point at which the curve crosses the y-axis and is given by the value of c in the general quadratic ax^2 + bx +c=0, which in this case is -3.

Once you have found these values, you can sketch the graph, see below.

Note: This graph was done by a computer, but a “sketch” doesn’t have to be perfect, it just has to be the right shape and cross the axes at the right points.

Level 8-9GCSEAQAEdexcelOCRWJEC

Example Questions

The quadratic on the left hand side of the equation factorises so that,

 

p^2-3p-10=(p+2)(p-5)

 

Therefore, we can rewrite the equation as

 

(p+2)(p-5)=0

 

For the left-hand side to be zero we require one of the brackets to be zero, hence, the two solutions to this quadratic equation are,

 

p=-2 and p=5

The quadratic on the left hand side of the equation factorises so that,

 

x^2-8x+15=(x-5)(x-3)

 

Therefore, we can rewrite the equation as

 

(x-5)(x-3)=0

 

For the left-hand side to be zero we require one of the brackets to be zero, hence, the two solutions to this quadratic equation are,

 

x=5 and x=3

To find the roots of a quadratic, we must set it equal to zero and find the solutions of that equation,

 

x^2-6x+8=0

 

Now, we must factorise. Observing that (-2)\times(-4)=8 and -2+(-4)=-6, we get that this quadratic factorises to

 

(x-2)(x-4)=0

 

So, for the left-hand side to be zero we require one of the brackets to be zero.

 

Therefore, the two roots of this quadratic equation are

 

x=2  and  x=4

The quadratic on the left hand side of the equation factorises so that,

 

2x^2+13x+15=(2x+3)(x+5)=0

 

For the left-hand side to be zero we require one of the brackets to be zero, hence, the two solutions to this quadratic equation are,

 

x=-\dfrac{3}{2} and x=-5

The quadratic on the left hand side of the equation factorises so that,

 

(3k-1)(k+6)=0

 

So, for the left-hand side to be zero we require one of the brackets to be zero. If the first bracket is zero, then we get

 

3k-1=0

 

If we add 1 to both sides and then divide by 3, we get the solution

 

k=\dfrac{1}{3}

 

If the second bracket is zero, then we get

 

k=-6

 

Therefore, the two solutions to this quadratic equation are,

 

k=\frac{1}{3} and k=-6

Related Topics

MME

Factorising Quadratics

Level 4-5Level 6-7GCSE
MME

Solving Equations

Level 4-5GCSEKS3

Worksheet and Example Questions

Site Logo

(NEW) Solving Quadratics By Factorisation Exam Style Questions - MME

Level 4-5 GCSENewOfficial MME

Drill Questions

Site Logo

Solving Quadratics By Factorising - Drill Questions

Level 4-5 GCSE
Site Logo

Solving Quadratics By Factorising (2) - Drill Questions

Level 4-5 GCSE

You May Also Like...

GCSE Maths Revision Cards

Revise for your GCSE maths exam using the most comprehensive maths revision cards available. These GCSE Maths revision cards are relevant for all major exam boards including AQA, OCR, Edexcel and WJEC.

£8.99
View Product

GCSE Maths Revision Guide

The MME GCSE maths revision guide covers the entire GCSE maths course with easy to understand examples, explanations and plenty of exam style questions. We also provide a separate answer book to make checking your answers easier!

From: £14.99
View Product