Number of Solutions Discriminant

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Johnny Yi 5 years, 11 months ago

Created this as a copy of "Impossible" questions on quadratic equations.

Name	Status	Author	Last Modified
"Impossible" questions on quadratic equations	Needs to be tested	Christian Lawson-Perfect	14/06/2017 16:17
Number of Solutions Discriminant	draft	Johnny Yi	30/04/2019 03:35

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Question statement

These questions might look impossible, but they're not: you can find the answers without doing any long calculations.

Give any introductory information the student needs.

			Name	Type	Generated Value

num_roots	integer	0
a	integer	1
b	integer	246
c	integer	11927365
d	integer	-16908
c1	integer	82253
c2	integer	34219

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Data type

Value

xxxxxxxxxx
 
random(0,1,2)

JME function reference

Description

Number of roots

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

→ Used by:

Advice
"Part b)" - Minimum accepted value
"Part b)" - Maximum accepted value

Advice

a)

Rearrange the equation:

$\simplify{x^2 = {-d}}$

Any number squared is positive, so this equation has {if(d>0,'no','two')} solutions.

b)

The discriminant of a quadratic equation show how many roots the equation has.

Look at the quadratic formula:

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

The value of $\sqrt{b^2-4ac}$ is called the discriminant, and the number of solutions to the equation depends on whether it is positive, negative, or zero:

If the discriminant is positive, there are two solutions to the equation.
If the discriminant is zero, the equation only has one solution.
If the discriminant is negative, the equation has no real solutions.

In this case, $a = 1$ , $b = \var{b}$ and $c = \var{c}$ .

Since $a = 1$ , the discriminant can be written $\sqrt{b^2-4c}$ .

We can tell that $b^2$ will have $\var{2*floor(log(b,10))+if(b/10^floor(log(b,10))>=sqrt(10),2,1)}$ digits. $c = \var{c}$ has $\var{ceil(log(c,10))}$ digits.

So, $b^2-4c$ is clearly {switch(num_roots=2,'positive',num_roots=0,'negative','zero')}, so the equation has {num_roots} real {pluralise(num_roots,'root','roots')}.

c)

In a quadratic equation $x^2+bx+c$ , $-b$ is the sum of the two roots of the equation and $c$ is their product.

So, we first need to find a pair of numbers which sum to $\var{c1+c2}$ . In fact, there are three pairs:

$\var{mod(c1,100)}$ and $\var{c1+c2-mod(c1,100)}$ ;
$\var{c1-mod(c1,10)}$ and $\var{c2+mod(c1,10)}$ ;
$\var{c1}$ and $\var{c2}$ .

The pair whose product is equal to $\var{c1*c2}$ are the solutions to the equation. We can work out which pair it is without doing the (very long!) multiplication.

$\var{mod(c1,100)} \times \var{c1+c2-mod(c1,100)}$ will not have as many digits as $\var{c1*c2}$ , so we can reject it.
$\var{c1-mod(c1,10)}$ and $\var{c2+mod(c1,10)}$ are both even, while $\var{c1*c2}$ is odd.
So $\var{c1}$ and $\var{c2}$ must be the solutions.

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This question is used in the following exam:

Andrew's copy of Discriminant by Andrew Siu in Andrew's workspace.

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Johnny Yi 5 years, 11 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Number entry

Advanced settings

a)

b)

c)

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Number of Solutions Discriminant

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Johnny Yi 5 years, 11 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Number entry

Advanced settings

a)

b)

c)

Generated value: `integer`