Complete the square and find solutions

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From users who are members of Transition to university :

Christian Lawson-Perfect

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Ready to use

7 years, 9 months ago

History

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Chris Graham commented 6 years, 7 months ago

Changed gaps to number entry, as they were not being marked correctly as mathematical expressions.

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Christian Lawson-Perfect 7 years, 9 months ago

Published this.

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Christian Lawson-Perfect 7 years, 9 months ago

Saved a checkpoint:

Extracted this from a bigger question on completing the square, and changed the variables so the roots are always nice integers.

Restore this checkpoint

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Christian Lawson-Perfect 7 years, 9 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 9 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.

Give any introductory information the student needs.

			Name	Type	Generated Value

big	number	-39
sml	number	10
bits	list	[ 5, 8 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
bits[0]^2-bits[1]^2

JME function reference

Description

The constant term in the expanded quadratic.

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

Can an exam override the value of this variable?

Generated value: `number`

-39

← Depends on:

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 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.

Write the following expression in the form $a(x+b)^2-c$ .

$\simplify {x^2+{sml}x+{big}} =$ 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)

Rewrite $x^2+\var{sml}x$ as $\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$ .

$\begin{align} \simplify[basic]{x^2+{sml}x+{big}} &= \simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\ &= \simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \text{.} \end{align}$

b)

We showed above that

$\simplify[basic]{x^2+{sml}x+{big}} = 0$

is equivalent to

$\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \text{.}$

We can then rearrange this equation to solve for $x$ .

$\begin{align} \simplify{(x+{bits[0]})^2-{(bits[1])^2} } &= 0 \\ (x+\var{bits[0]})^2 &= \var{bits[1]^2} \\ x+\var{bits[0]} &= \pm \var{bits[1]} \\ x &= -\var{bits[0]} \pm \var{bits[1]} \\[2em] x_1 &= \var{-bits[0]-bits[1]} \text{,}\\ x_2 &= \var{-bits[0]+bits[1]} \text{.} \end{align}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Chris Graham commented 6 years, 7 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

Parts

Gap-fill

a)

b)

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Complete the square and find solutions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Chris Graham commented 6 years, 7 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

a)

b)

Generated value: `number`