Solution of Quadratic Equation

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Question statement

A new way for solving Quadratic Equation:

The quadratic equation $ax^2+bx+c=0$ can be solved in a different and easy way as follows:

Let $\alpha$ and $\beta$ be the two roots. Then we know $\alpha\times\beta=\dfrac{c}{a}$ and $\alpha+\beta=-\dfrac{b}{a}$

We shall use the property that the roots are equidistant from their average.

So we can take them as $\alpha=\dfrac{1}{2}\rm{sum}-u$ and $\beta=\dfrac{1}{2}\rm{sum}+u$ Or $\boxed{\alpha=-\dfrac{1}{2}\dfrac{b}{a}-u}$ and $\boxed{\beta=-\dfrac{1}{2}\dfrac{b}{a}+u}$ .

Consequently we have $\alpha\times\beta=\left(-\dfrac{1}{2}\dfrac{b}{a}-u\right)\times\left(-\dfrac{1}{2}\dfrac{b}{a}+u\right)$

$\implies \dfrac{c}{a}=\dfrac{1}{4}\dfrac{b^2}{a^2}-u^2\implies u^2=\dfrac{1}{4}\dfrac{b^2}{a^2}-\dfrac{c}{a}$ .

Simplifying $u^2=\dfrac{b^2-4ac}{4a^2}\implies \boxed{u=\dfrac{\sqrt{b^2-4ac}}{2a}}$ .

Substituting above we get

$\alpha=\dfrac{-b-\sqrt{b^2-4ac}}{2a}$ and $\beta=\dfrac{-b+\sqrt{b^2-4ac}}{2a}$ .

Note : $\boxed{\alpha=\rm{average}-u}$ and $\boxed{\beta=\rm{average}+u}$

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Let us consider the following example:

$x^2-5x+6=0$ .

We generally start from the product $6$ and find factors by inspection so that the sum will be $5$ . In stead, in this method we start from the sum and find $u$ .

Now sum of roots $=5\implies \rm{average}=\dfrac{5}{2}$ .

So roots are $\dfrac{5}{2}-u$ and $\dfrac{5}{2}+u$ .

Therefore product of roots $=\dfrac{25}{4}-u^2\implies 6=\dfrac{25}{4}-u^2\implies u^2=\dfrac{25}{4}-6\implies u^2=\dfrac{1}{4}\implies u=\pm\dfrac{1}{2}$ .

So roots are $\dfrac{5}{2}-\dfrac{1}{2}$ and $\dfrac{5}{2}+\dfrac{1}{2}$ . Or roots are $2$ and $3$ .

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Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	1
n	integer	7
b	integer	8
c	integer	7
dis	number	36
sum	list	[ -8, 1 ]
product	list	[ 7, 1 ]
avg	list	[ -4, 1 ]
u	list	[ 3, 1 ]
alpha	list	[ -7, 1 ]
beta	list	[ -1, 1 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..5)*random(1,-1)

JME function reference

Description

First coefficient

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

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

b
dis
product
sum
u

"Unnamed part" - prompt
"Unnamed part" → "Gap 0." - Minimum accepted value
"Unnamed part" → "Gap 0." - Maximum accepted value
"Unnamed part" → "Gap 6." - Minimum accepted value
"Unnamed part" → "Gap 6." - Maximum accepted value

Parts

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1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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10. Gap Gap 9. Number entry
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11. Gap Gap 10. Number entry
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Gap-fill

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

PRACTICE EXERCISE:

Solve the equation $\simplify{{a}x^2+{b}x+{c}=0}$

Fill in the gaps below.

a=Gap 0.

b=Gap 1.

c=Gap 2.

Sum of roots=Gap 3.

Avg(Average of roots)=Gap 4.

$b^2$ =Gap 5.

$4ac$ =Gap 6.

Discriminant $(b^2-4ac)$ =Gap 7.

$u\left(=\dfrac{\sqrt{b^2-4ac}}{2a}\right)$ =Gap 8.

$\alpha(=Avg-u)$ =Gap 9.

$\beta(=Avg+u)$ =Gap 10.

Advice

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

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This question is used in the following exam:

My First Exam by Jamie Coventry in Jamie's workspace.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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AJAY OTTA 4 years, 10 months ago

AJAY OTTA 4 years, 10 months ago

AJAY OTTA 4 years, 11 months ago

AJAY OTTA 5 years ago

AJAY OTTA 5 years, 1 month ago

AJAY OTTA 5 years, 1 month ago

A new way for solving Quadratic Equation:

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

PRACTICE EXERCISE:

Solve the equation $\simplify{{a}x^2+{b}x+{c}=0}$

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Solution of Quadratic Equation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

AJAY OTTA 4 years, 10 months ago

AJAY OTTA 4 years, 10 months ago

AJAY OTTA 4 years, 11 months ago

AJAY OTTA 5 years ago

AJAY OTTA 5 years, 1 month ago

AJAY OTTA 5 years, 1 month ago

A new way for solving Quadratic Equation:

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

PRACTICE EXERCISE:

Solve the equation {{a}x2+{b}x+{c}=0}\simplify{{a}x^2+{b}x+{c}=0}

Generated value: `integer`

Solve the equation $\simplify{{a}x^2+{b}x+{c}=0}$