Simon's copy of Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1

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Simon Thomas 6 years, 1 month ago

Created this as a copy of Factorising Quadratic Equations with

Name	Status	Author	Last Modified
Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1	Ready to use	Hannah Aldous	20/11/2019 14:39
MATH6058 Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1	draft	Catherine Palmer	25/01/2018 09:18
cormac's copy of MATH6058 Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1	draft	cormac breen	24/10/2018 13:36
Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1	draft	Ed Southwood	08/08/2022 10:18
Factorising Quadratic Equations with $x^2$ Coefficients Less than 1	draft	Ed Southwood	08/08/2022 10:18
Simon's copy of Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1	draft	Simon Thomas	07/03/2019 09:41
AD4 Quadratics - factorise (a not 1)	Ready to use	Andrew Neate	09/04/2024 15:14

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
b	integer	1
c	integer	3
d	integer	1
roots	list	[ -1/2, -1/3 ]

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..3)

JME function reference

Description

$a$ in $(ax+b)(cx+d)$

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

→ Used by:

c
roots

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Factorise the equation

$\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}\text{.}$

$(\var{a}x+\phantom{.}$ Gap 0. $) ($ Gap 1. $x+\phantom{.}$ Gap 2. $)\; = 0$

Advice

(a)

This question is more difficult because we are factorising a quadratic of the form $ax^2+bx+c$ where $a \neq 1$ .

To factorise an expression like $ax^2+bx+c$ we can start by finding two numbers which multiply to give $a\times c$ and add to give $b$ .

In our example, to factorise $\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}$ we start by finding two numbers which:

Multiply to give $\var{a*b*c*d}$ $(a \times c)$ and add to give $\simplify{{a}*{d}+{b}*{c}}$ $(b)$

Note that $(\var{a*d}) \times (\var{b*c}) = \var{a*b*c*d}$

and

( $\var{a*d})+(\var{b*c}) = \var{a*d+b*c}$

So the two numbers required are $\var{a*d}$ and $\var{b*c}$ .

We can then return to our original quadratic and split the $x$ coefficient into two pieces, $\var{a*d}x$ and $\var{b*c}x$ , as follows:

$\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}}= \simplify[!cancelterms]{{a*c}x^2+{a*d}x+{b*c}x+{b*d}}$

Now we can factorise the first two terms and the last two terms separately as:

$\simplify[!cancelterms]{{a*c}x^2+{a*d}x+{b*c}x+{b*d}} = \simplify[!cancelterms]{{a}x({c}x+{d})+{b}({c}x+{d})}$

which is the same as

$\simplify[!cancelterms]{({a}x+{b})({c}x+{d})}$ which is thus our final factorisation.

(b)

$\simplify{({a}x+{b})({c}x+{d}) = 0}$ when either $\simplify{{a}x+{b}} = 0$ or $\simplify{{c}x+ {d}} = 0$ .

So the roots of the equation are $\var[fractionnumbers]{-b/a}$ and $\var[fractionnumbers]{-d/c}$ .

Give a worked solution to the whole question.

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This question is used in the following exam:

Quadratic expressions and equations by Simon Thomas in Algebra.

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Simon's copy of Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Simon Thomas 6 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

(a)

(b)

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Simon's copy of Factorising Quadratic Equations with x2x^2 Coefficients Greater than 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

(a)

(b)

Simon's copy of Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1

Generated value: `integer`