Rafa?'s copy of Solving a quadratic equation

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Rafa? Porzezinski

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Rafa? Porzezinski 7 years, 3 months ago

Created this as a copy of Solving a quadratic equation.

Name	Status	Author	Last Modified
Solving equations	draft	Bill Foster	06/08/2020 08:41
Solving a quadratic equation	draft	Lois Rollings	29/07/2016 13:42
Francis 's copy of Solving a quadratic equation	draft	Francis Duah	12/08/2016 17:55
Rafa?'s copy of Solving a quadratic equation	draft	Rafa? Porzezinski	11/01/2018 14:14

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Question statement

Znajdź pierwiastki podanego równania kwadratowego

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	-5
b	integer	1
d	integer	-3
f	integer	2
s3	integer	-1
s2	integer	1
s1	integer	1
n4	number	1
n2	integer	1
disc	number	1
rdis	string	The roots are real and distinc
n1	integer	-11
c1	integer	5
n3	integer	4
rep	string
n5	integer	2
d1	integer	3

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random(2..5)

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Description

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

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.

$\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}=0$

Podja wartości pierwiastków jako ułamki lub liczby całkowite, nie dziesiętne

Wpisz pierwiastek z niższą wartością jako pierwszy, jeżeli równanie posiada jeden podwójny pierwiastek wpisz go w obu miejscach.

Pierwiuastek z niższą wartością $x=\;$ Gap 0..

Pierwiastek z wyższa wartością $x=\;$ Gap 1.

Advice

There are different ways to solve a quadratic equation.

If you can see how to factorise the quadratic expression then this should be the quickest method.

For this example we have

$\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}$

So either

$\simplify{ ({a} * x + { -c}) =0} \ \text{or} \ \ \simplify{ ({b} * x + { -d})}=0$

So the roots are:

$x= \simplify{{n1-n4}/{2*a*b}} \ \ \text{and} \ \ x= \simplify{{n1+n4}/{2*a*b}}$

If you cannot see a factorisation then you can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Remember that there are three possible types of solution, depending upon the value of the discriminant $\Delta=b^2-4ac$ .

1. $\Delta \gt 0$ . The roots are real and distinct

2. $\Delta=0$ . The roots are real and equal. Their common value is $\displaystyle -\frac{b}{2a}$

3. $\Delta \lt 0$ . There are no real roots. (The roots are complex)

For this question the discriminant is $\Delta = \simplify[std]{{-n1}^2-4*{a*b*c*d}}=\var{disc}$ . {rdis}

So the {rep} roots are:

$x = \frac{\var{n1} - \sqrt{\var{disc}}}{\var{n3}} = \frac{\var{n1} - \var{n4} }{\var{n3}} = \simplify{{n1 - n4}/ {n3}} \ \ \text{and} \ \ x = \frac{\var{n1} + \sqrt{\var{disc}}}{\var{n3}} = \frac{\var{n1} + \var{n4} }{\var{n3}} = \simplify{{n1 + n4}/ {n3}}$

You could also use the method of completing the square.

First we write:

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

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

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

See also these Mathcentre leaflets on solving quadratic equations.

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Rafa?'s copy of Solving a quadratic equation

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Rafa? Porzezinski 7 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

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Rafa?'s copy of Solving a quadratic equation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Rafa? Porzezinski 7 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`