Complete the square

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Created this.

Name	Status	Author	Last Modified
Complete the square	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Kevin's copy of Complete the square	draft	Kevin Bohan	05/06/2019 10:28

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Question statement

Complete the square for the quadratic expression $q(x)$ by writing it in the form

$a(x+b)^2+c$ for numbers

$a,\;b$ and

$c$ .

Hence find both roots of the equation $q(x)=0$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	-1
b	integer	1
d	integer	-5
f	integer	2
s3	integer	-1
s2	integer	-1
s1	integer	-1
n4	number	9
n2	integer	9
disc	number	81
rdis	string	The roots exist and are distin
n1	integer	-11
rep	string
n3	integer	4
c1	integer	1
n5	integer	2
d1	integer	5

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Gap-fill

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

$q(x)=\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}$
Now write

$q(x) = a(x+b)^2+c\;\;$ for fractions or integers

$a$ ,

$b$ ,

$c$ .

$q(x)=\;$ Unnamed gap

You can get more information on completing the square by clicking on Show steps.

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Remember: Input all numbers as fractions or integers and not as decimals.

Advice

Completing the square for the quadratic expression $\simplify{{a*b}x^2+{-n1}x+{c*d}}$ .

$\begin{eqnarray} \simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\var{n5}\left(\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\right)\\ &=&\var{n5}\left(\left(\simplify{x+({-n1}/{2*a*b})}\right)^2+ \simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\right)\\ &=&\var{n5}\left(\left(\simplify{x+({-n1}/{2*a*b})}\right)^2 -\simplify{ {n2^2}/{4*(a*b)^2}}\right)\\ &=&\var{n5}\left(\simplify{x+({-n1}/{2*n5})}\right)^2 -\simplify{ {n2^2}/{4*(n5)}} \end{eqnarray}$
So to solve

$\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:

$\begin{eqnarray} \left(\simplify{x+({-n1}/{2*a*b})}\right)^2&\phantom{{}}& -\simplify{ {n2^2}/{4*(a*b)^2}}=0\Rightarrow\\ \left(\simplify{x+({-n1}/{2*a*b})}\right)^2&\phantom{{}}&=\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \end{eqnarray}$
So we get the two {rep} solutions:

$\begin{eqnarray} \simplify{x+({-n1}/{2*a*b})}&=&\simplify{{abs(n2)}/{2*a*b}} \Rightarrow &x& = \simplify{({abs(n2)+n1}/{2*a*b})}\\ \simplify{x+({-n1}/{2*a*b})}&=&\simplify{-({abs(n2)}/{2*a*b})} \Rightarrow &x& = \simplify{({n1-abs(n2)}/{2*a*b})} \end{eqnarray}$

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

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Complete the square

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`