Eliminating a square root

[jpanknin]

I'm having trouble understanding a specific step in the following problem. The original problem is:



I've simplified to:



And this is where I get lost. I've squared each term individually as in the following:



whereas the guide I'm using does the following:



This version squares the (6540 - 2d) as a single term getting (42,771,600 - 26,160d + 4d²) rather than squaring the 6540 and the 2d separately getting 42,771,600 and 4d².

Can anyone explain why the (6450 - 2d)² is the proper way vs. squaring each term separately (i.e. 6450² and 2d²)? Thank you

 

[Deleted member 4993]

I'm having trouble understanding a specific step in the following problem. The original problem is:

View attachment 16547

I've simplified to:

View attachment 16548

And this is where I get lost. I've squared each term individually as in the following:

View attachment 16551

whereas the guide I'm using does the following:

View attachment 16549

This version squares the (6540 - 2d) as a single term getting (42,771,600 - 26,160d + 4d²) rather than squaring the 6540 and the 2d separately getting 42,771,600 and 4d².

Can anyone explain why the (6450 - 2d)² is the proper way vs. squaring each term separately (i.e. 6450² and 2d²)? Thank you

You said:

I've squared each term individually as in the following:

You CANNOT do that (legally). You have

6540 = 2*d + 545 * d^1/2

6540 - 2*d = 545 * d^1/2 ..... Now square each side

[6540 - 2*d]² = 545² * d

Upon simplification, this will give you a quadratic equation. You can solve that by your favorite method.

 

[jpanknin]

Thank you, but I still don't understand WHY it must be done this way. Any proofs or examples as to the underlying reasons?

 

[Deleted member 4993]

I'm having trouble understanding a specific step in the following problem. The original problem is:

View attachment 16547

I've simplified to:

View attachment 16548

And this is where I get lost. I've squared each term individually as in the following:

View attachment 16551

whereas the guide I'm using does the following:

View attachment 16549

This version squares the (6540 - 2d) as a single term getting (42,771,600 - 26,160d + 4d²) rather than squaring the 6540 and the 2d separately getting 42,771,600 and 4d².

Can anyone explain why the (6450 - 2d)² is the proper way vs. squaring each term separately (i.e. 6450² and 2d²)? Thank you

The mathematical law is - you have to "operate" similarly to both sides as a group (not operate on "each term").

Here the objective is to get rid of the "square root". That is why you have "correctly" chose to use "squaring" operation. But you have to do it "legally" and "judiciously".

You could "legally" do:

6540² = [545d^1/2 +2d]²

But that will still leave d^1/2.

Thus you should do:

[6540 - 2d]² = [545d^1/2]²

Upon simplification - d^1/2 is gone!!!

 

[pka]

Thank you, but I still don't understand WHY it must be done this way. Any proofs or examples as to the underlying reasons?

You are simply mistaken. \((a+b)^2\ne a^2+b^2\)
Rather \((a+b)^2=a^2+2ab+b^2\).
Thus \((545\sqrt d+2d)^2=545^2d+2(545)(\sqrt d)(2d)+4d^2\)

 

[Dr.Peterson]

Thank you, but I still don't understand WHY it must be done this way. Any proofs or examples as to the underlying reasons?

When we say "do the same thing to both sides", we mean, "apply the same operation to the value of each entire side". So when you square both sides, that has to mean squaring the entire quantity, not each term separately.

 

[Steven G]

Clearly 3+2 = 5.

The question is if 3^2 + 2^2 = 5^2. That is does 9+4 =25 or does 13=25. The answer is NO.

However since 3+2 = 5 we can say that (3+2)^2 =5^2. This is because both the left hand side and the right side equal 25.

 

[jpanknin]

Thank you very much. That clears it up very well. Appreciate everyone's time and responses.