Completing the square

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Adrian Jannetta 6 years, 5 months ago

Created this as a copy of Completing the square.

Name	Status	Author	Last Modified
Completing the square	Ready to use	Hannah Aldous	20/11/2019 14:34
Complete the square and find solutions	Ready to use	Christian Lawson-Perfect	20/11/2019 14:34
Inbbavathie's copy of Complete the square and find solutions (SP1)	draft	Inbbavathie Ravi	15/07/2017 22:57
Inbbavathie's copy of Complete the square and find solutions	draft	Inbbavathie Ravi	24/07/2017 03:48
Katherine's copy of Complete the square and find solutions	draft	Katherine Tomlinson	02/08/2017 12:47
Complete the square and find solutions [L4 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 10:17
Ann's copy of Complete the square and find solutions	draft	Ann Smith	09/10/2018 12:27
Completing the square	draft	Adrian Jannetta	16/08/2023 14:56
Complete the square and find solutions	draft	Johnny Yi	02/12/2020 01:07
Simon's copy of Completing the square	draft	Simon Thomas	07/03/2019 10:08
Simon's copy of Complete the square and find solutions	draft	Simon Thomas	07/03/2019 10:10
JD's copy of Completing the square	draft	JD Ichwan	21/05/2019 05:37
Completing the square and finding roots of a quadratic equation	draft	Maria Pickett	11/09/2019 13:51
Andrew's copy of Complete the square and find solutions	draft	Andrew McKinley	25/09/2019 13:38
AD3 Completing the square	Ready to use	Andrew Neate	09/04/2024 15:14

There are 31 other versions that do you not have access to.

Question statement

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called "completing the square".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

Rewrite the following expressions in the form

$(x+b)^2-c$ or

$a(x+b)^2-c$

Give any introductory information the student needs.

			Name	Type	Generated Value

evens1	integer	30
evens2	integer	16
evens3	integer	20
odds	integer	25
odds2	integer	27
odds3	integer	21

			Name	Type	Generated Value

all	integer	5
all2	integer	2
multiall	integer	40
big	integer	39
sml	integer	6
multiall2	integer	20

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(10..30 #2)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

evens2
evens3

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  Add
Part c) Gap-fill
1. Gap Mathematical expression
  Add
Part d) Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

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

$x^2+ \var{evens1}x =$ Unnamed gap

Advice

Completing the square works by noticing that

$(x+a)^2 = x^2 + 2ax + a^2$

So when we see an expression of the form $x^2 + 2ax$ , we can rewrite it as $(x+a)^2-a^2$ .

a)

We have $x^2+ \var{evens1}x$ , so we can replace it with $(x+\var{evens1/2})^2-\var{evens1/2}^2 = (x+\var{evens1/2})^2 - \var{evens1^2/4}$ .

Check that this is equivalent to the original expression by expanding the brackets:

$\begin{align} (x+\var{evens1/2})^2 - \var{evens1^2/4} &= \simplify[basic]{ x^2 + 2*{evens1/2}*x + {evens1/2}^2 - {evens1^2/4} } \\ &= x^2 + \var{evens1}x \text{.} \end{align}$

b)

Replace $x^2 + \var{odds}x$ by $\simplify[basic]{(x+{odds}/2)^2-({odds}/2)^2}$ , to obtain

$\begin{align} x^2 + \var{odds}x &= \simplify[basic]{(x+{odds}/2)^2-({odds}/2)^2} \\[0.5em] &= \simplify[basic]{ (x+{odds}/2)^2 - {odds^2}/4} \text{.} \end{align}$

c)

Replace $x^2+\var{evens2}x$ with $(x+\var{evens2/2})^2 - \var{evens2/2}^2$ . Remember to keep the $\var{evens2-evens1}$ term on the end!

$\begin{align} \simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} &= \simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\ &= \simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} } \end{align}$

d)

First, notice that $\simplify[basic]{ {all}x^2 + {multiall}x } = \simplify[basic]{ {all}*( x^2 + {multiall/all} x)}$ .

Then, we can replace $x^2 + \var{multiall/all}x$ with $(x+\var{multiall/all/2})^2 - \var{multiall/all/2}^2$ .

$\begin{align} \simplify[basic]{ {all}x^2 + {multiall}x + {odds3-evens2}} &= \simplify[basic]{ {all}*( x^2 + {multiall/all} x) + {odds3-evens2}} & \text{Extract the common factor of } \var{all} \\ &= \simplify[basic]{ {all}*( (x+{multiall/all/2})^2 - {multiall/all/2}^2) + {odds3-evens2} } & \text{Complete the square}\\ &= \simplify[basic]{ {all}*(x+{multiall/all/2})^2 - {all}*{(multiall/all/2)^2} + {odds3-evens2} } & \text{Expand the constant term}\\ &= \simplify[basic]{ {all}*(x+{multiall/all/2})^2 + {odds3-evens2 - (multiall/2)^2/all}} & \text{Collect constants} \end{align}$

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Taxonomy: mathcentre

Taxonomy: Kind of activity

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Checkpoint description

Adrian Jannetta 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

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Error

Completing the square

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Adrian Jannetta 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`