an apparently arcane question: using Quadratic Formula when b^2 is large in comparison to ac

[allegansveritatem]

I have been all over the internet looking for info on this subject and found nothing. A couple of sites mentioned it but the explanations were sketchy and/or downright obscure. Here is the problem:

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When computations are carried out on a calculator, the Quadratic Formula will not always give accurate results if b² is large in comparison to ac, because one of the roots will be close to zero and difficult to approximate.

(a) Use the Quadratic Formula to approximate the roots of the given equations.
(b) To obtain a better approximation for the root near zero, rationalize the numerator to change

. . . . .[math]x\, =\, \dfrac{-b\, \pm\, \sqrt{b^2\, -\, 4ac\,}}{2a}\, \mbox{ to }\, x\, =\, \dfrac{2c}{-b\, \mp\, \sqrt{b^2\, -\, 4ac\,}}[/math]
[math]43.\quad x^2\, +\, 4,500,000x\, -\, 0.96\, =\, 0[/math]
[math]44.\quad x^2\, -\, 73,000,000x\, +\, 2.01\, =\, 0[/math]

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What I want to know here is not how to calculate the figures, but how did they derive that alternate form of the equation. The excellent solutions manual I have goes through the whole process of derivation but does not go into enough detail to make it clear. What is exercising my curiosity to the max is: How to multiply the conjugate of a term containing+/- with its conjugate and what does it mean when you flip the sign to -/+. Here is what the solutions manual has to say:


so, I guess my question is this: Why flip the sign and what does that effect? And my other question has to do with the words:"The root near zero was obtained in part (a) using the plus sign. In the second formula it corresponds to the minus sign." Say what???

 

[Steven G]

Suppose you have a +/- b. Now as I am sure you know, a +/- b is shorthand notation for a + b and a - b.

Now the conjugate of a + b is a-b and the conjugate for a - b is a + b. Note that the + and - signs get switched in the conjugate. So one can write that the conjugate of a +/- b is a -/+ b. When reading, the rule is to use BOTH top signs or Both bottoms but NEVER use one top sign and one bottom sign.
Also if you have - (+/- b) = -/+ b. Actually they are the same, but we use -/+ just to show that you multiplied by a negative number.

 

[Dr.Peterson]

To understand this better, just take it one sign at a time. Take the positive sign on the radical and rationalize the denominator; you'll see a negative sign in the end. Then repeat with a negative sign, and the result will have a positive sign.

The sign ∓ is used where ± is already in use, to indicate that the sign is to be taken as opposite - that is, when one is read as +, the other is read as -. The notation is just a shorthand allowing you to combine two sets of work into one; you can always do it one sign at a time to clarify. With experience, you will become able to read this more easily.

Be sure to calculate both versions by hand to get a better sense of why they did what they did.

 

[allegansveritatem]

Suppose you have a +/- b. Now as I am sure you know, a +/- b is shorthand notation for a + b and a - b.

Now the conjugate of a + b is a-b and the conjugate for a - b is a + b. Note that the + and - signs get switched in the conjugate. So one can write that the conjugate of a +/- b is a -/+ b. When reading, the rule is to use BOTH top signs or Both bottoms but NEVER use one top sign and one bottom sign.
Also if you have - (+/- b) = -/+ b. Actually they are the same, but we use -/+ just to show that you multiplied by a negative number.

 

[allegansveritatem]

I begin to see. We are doing two operations in one go? What do you mean by this: "When reading use both top signs..." When reading as opposed to what? When doing the operations?

 

[allegansveritatem]

To understand this better, just take it one sign at a time. Take the positive sign on the radical and rationalize the denominator; you'll see a negative sign in the end. Then repeat with a negative sign, and the result will have a positive sign.

The sign ∓ is used where ± is already in use, to indicate that the sign is to be taken as opposite - that is, when one is read as +, the other is read as -. The notation is just a shorthand allowing you to combine two sets of work into one; you can always do it one sign at a time to clarify. With experience, you will become able to read this more easily.

Be sure to calculate both versions by hand to get a better sense of why they did what they did.

So we are dealing with a kind of double operation here? Of course. Yes The plus/minus would indicate something like that. Two things. I will do as you suggest and work it out in the morning one at a time. Thanks.

 

[allegansveritatem]

thanks to all who responded. I worked this out in two stages and now it is pretty clear to me what is going on.