Solving a quadratic equation

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Lois Rollings

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Lois Rollings 8 years, 7 months ago

Published this.

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Lois Rollings 9 years, 7 months ago

Created this as a copy of Solving equations.

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

Find the roots of the following quadratic equation.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	1
b	integer	3
d	integer	1
f	integer	6
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	5
c1	integer	1
n3	integer	12
rep	string
n5	integer	6
d1	integer	1

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Description

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Generated value: `integer`

→ Used by:

d
disc
f
n1
n2
n3
n5

Advice
"Unnamed part" - prompt
"Unnamed part" → "Gap 0." - Correct answer
"Unnamed part" → "Gap 1." - Correct answer

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

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

Enter the values of the roots as fractions or integers, not as decimals.

Enter the root with the lower value first. If the equation has two equal roots then enter this value in both input boxes.

The root with the lower value is $x=\;$ Gap 0.. The root with the higher value is $x=\;$ 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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Lois Rollings 8 years, 7 months ago

Lois Rollings 9 years, 7 months ago

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Generated value: `integer`

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Solving a quadratic equation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Lois Rollings 8 years, 7 months ago

Lois Rollings 9 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`