cormac's copy of Solve quadratic inequalities

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cormac breen 6 years, 6 months ago

Created this as a copy of Solve quadratic inequalities.

Name	Status	Author	Last Modified
Solving linear inequalities	Ready to use	Bradley Bush	20/11/2019 14:42
Solve quadratic inequalities	Ready to use	Bradley Bush	20/11/2019 14:42
Solving linear inequalities	Should not be used	Adrian Jannetta	31/08/2023 17:00
Ida's copy of Solve quadratic inequalities	draft	Ida Landgärds	13/04/2018 09:16
Ida's copy of Solving linear inequalities	draft	Ida Landgärds	13/04/2018 09:16
Solving linear inequalities [L5 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 10:51
cormac's copy of Solving linear inequalities	draft	cormac breen	24/10/2018 13:44
cormac's copy of Solve quadratic inequalities	draft	cormac breen	24/10/2018 13:46
Solving linear inequalities	draft	Xiaodan Leng	11/07/2019 01:43

There are 66 other versions that do you not have access to.

Question statement

Solve the following quadratic inequalities by firstly factorising $f(x)$ and then solving for $x$ when $f(x)=0$. It may be helpful to sketch each quadratic.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
b	integer	5
c	integer	6
d	integer	3
g	integer	4
f	integer	8

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Value

xxxxxxxxxx
 
random(1..5)

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Description

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Generated value: `integer`

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Parts

Gap-fill

Gap-fill

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Find the range of vaues such that: $\simplify{({g}+{f})x-{g}{f}>x^2}$

Rearrange then factorise the inequality:

Gap 0. $<0$.

Use the above result to find the range of values for $x$ such that $\simplify{({g}+{f})x-{g}{f}>x^2}$.

$x>$ Gap 1.

Gap 3.

$x<$ Gap 2.

Advice

a)

This example is best illustrated by the graph of $\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}$ below. By finding the roots of the equation we can find the $x$ coordinates where the line crosses the $x$ axis and then we can use a sketch or visualise the graph to work out the set of values for $x$ where $f(x)>0$.\[\\[0.1em]\]

{geogebra_applet('Hk5dTptY',[["a",a],["b",b]])}

\[
\begin{align}
\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}&>0\\
\simplify{(x+{a})(x-{b})}&=0\text{.}
\end{align}
\]

Therefore

\[ \simplify{x<{-a}}\text{ or }\simplify{x>{b}}\text{.}\]

b)

We use the same method in this example but this time we use our graph to visualise where $g(x)<0$.\[\\[0.1em]\]

{geogebra_applet('PUFStTNa',[["c",c],["d",d]])}

\[
\begin{align}
\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}&<0\\
\simplify{(x+{c})(x-{d})}&=0\text{.}
\end{align}
\]

Therefore

\[ \simplify{x>{-c}}\text{ and }\simplify{x<{d}}\text{.}\]

c)

The notable difference with solving this equation is the requirement to rearrange the inequality before factorisation.\[\\[0.1em]\]

{geogebra_applet('CZsSCdH6',[["e",g],["f",f]])}

\[
\begin{align}
\simplify{({g}+{f})x-{g}{f}}&>x^2\\
\simplify{x^2-({g}+{f})x+{g}{f}}&<0\\
\simplify{(x-{g})(x-{f})}&=0\text{.}
\end{align}
\]

Therefore

\[ \simplify{x>{g}}\text{ and }\simplify{x<{f}}\text{.}\]

Alternatively, we can plot the graph of $x^2$ against $\simplify{({g}+{f})x-{g}{f}}$ to visualise the same result.

{geogebra_applet('Ez697SNE',[["g",g],["f",f]])}

From this graph we can see that the values of $x$ where $\simplify{({g}+{f})x-{g}{f}}>x^2$ are the same as the values of $x$ where $\simplify{x^2-({g}+{f})x+{g}{f}}<0$.

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This question is used in the following exams:

MATH1901 CA V2 by cormac breen in cormac's workspace.
MATH1901 Quadratic Equations & Further Algebra by Blathnaid Sheridan in Blathnaid's workspace.
MATH1901 CA by cormac breen in cormac's workspace.

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cormac's copy of Solve quadratic inequalities

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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cormac breen 6 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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Error

cormac's copy of Solve quadratic inequalities

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

cormac breen 6 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`