Complete the square. (Video)

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Bill Foster

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Bill Foster 55 years, 4 months ago

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Question statement

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

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

$a,\;b$ and

$c$ .

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
c	integer	5
b	integer	3
d	integer	5
f	integer	12
s3	integer	1
s2	integer	1
s1	integer	1
n5	integer	12
disc	number	25
c1	integer	5
rep	string
n1	integer	35
n2	integer	-5
n3	integer	24
n4	number	5
rdis	string	The roots exist and are distin
d1	integer	5

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Generated value: `integer`

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d
disc
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n1
n2
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n5

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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.

$q(x)=\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}$
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. You will lose 1 mark if you do so.

You will also find a video covering the use of completing the square in solving a quadratic equation.

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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This question is used in the following exams:

mathcentre: Completing the square by Bill Foster in Bill's workspace.
Bradford: Completing the square by Bill Foster in Bill's workspace.
Maths Support: Completing the square by Christian Lawson-Perfect in Maths Support Wiki.
Solving equations: 1 by Aneela Arshad in Aneela's workspace.
Bradford: Diagnostic test - Basic maths by Bill Foster in Bill's workspace.
Basic Maths Revision by Gareth Woods in Gareth's workspace.
Peter's test exam by Peter Samuels in Peter's workspace.
Solving Quadratic Equations by Gareth Woods in Gareth's workspace.
mathcentre: Diagnostic test - Basic maths by Bill Foster in Bill's workspace.
Algebra: 1 by Aneela Arshad in Aneela's workspace.

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Taxonomy: mathcentre

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Bill Foster 55 years, 4 months ago

Error in variable testing condition

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Generated value: `integer`

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Complete the square. (Video)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`