Completing the Square: 2x^2 + 12x + 3 = 0

John Whitaker

Junior Member
Joined
May 9, 2006
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89
I have:

2x^2 + 12x + 3 = 0 (divide by 2)
x^2 + 6x + 3/2 = 0
x^2 + 6x = - 3/2
X^2 + 6x + 9 = 15/2
(x + 3 )^2 = 15/2
x + 3 = +/- sqrt (15/2)
x + 3 = +/- sqrt (30)/2

My question is: What arithmatical process determines that:
= +/- sqrt (15/2) should become +/- sqrt (30)/2

Please explain how the "30" comes in. Thank you.
 
Re: Completing the Square

John Whitaker said:
I have:

2x^2 + 12x + 3 = 0 (divide by 2)
x^2 + 6x + 3/2 = 0
x^2 + 6x = - 3/2
X^2 + 6x + 9 = 15/2
(x + 3 )^2 = 15/2
x + 3 = +/- sqrt (15/2)
x + 3 = +/- sqrt (30)/2

My question is: What arithmatical process determines that:
= +/- sqrt (15/2) should become +/- sqrt (30)/2

Please explain how the "30" comes in. Thank you.

In the days BC (before calculators) it became a general agreement that one should never leave a fraction under a radical sign, nor leave a radical in the denominator of a fraction. This simplified the arithmetic involved in doing computations.

So...you've got sqrt(15/2), with a fraction under the radical sign. And writing it as sqrt(15) / sqrt(2) doesn't help, either...that has a radical in the denominator of a fraction. BUT....suppose we wrote 15/2 as a fraction whose denominator was a perfect square? Multiply numerator and denominator of 15/2 by 2 to get 30/4. And write sqrt(30/4) as sqrt(30)/sqrt(4). And sqrt(4) is 2, so your sqrt(15/2) becomes sqrt(30) / 2!

I hope that answers your question....in the days before calculators, it was much easier to look up sqrt(30) in a table and divide it by 2 than it was to look up sqrt(15) and sqrt(2), and then divide 3.8730 by 1.4142. Of course, since calculators have come into play, that's not so important. One can easily find sqrt(15/2) on a calculator.
 
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