Basics math confusion, help me

[PsychoRebellious]

To start with, I'm a HSC failure who failed it and gave up on it because I couldn't stand learning something without knowing all the ins and outs of it. Questions like WHY we are following one method but not other offended me to the point that I started finding sciences dull and boring and left HSC and started learning computer programming. Anyway, after a few years of learning to code, I am jumping into cryptography and complex algorithms and I need to sharpen my maths skills now, and here I am studying everything again from online sources and books and this time I am going to make sure to not be left unsatisfied. Anyway, I was reading an HSC Mathematics books and here's one confusion in Surds and rationalizing their denominators
Consider the following:
(5x-16)/(√y - 10)

Now to solve it the book says you rationalize denominator by removing the surd from the denominator. The book continues by multiplying the above equation with (√y+10)/(√y+10) and in the process having y -100 as denominator. Now all that confuses me is that why not multiply the original statement/equation with
(√y-10) / (√y-10) and remove the surd by applying (a+b)^2 formula? So I tried it out and the two answers differ from each other. Anyway, so here's what i did. Took a value for x and y and used them in both the cases. Now let be elaborate
(* is the multiplication symbol)
Case 1:

Code: [Select]

(5x-16) / (√y-10)       * (√ y-10)/(√ y-10)

Plotting the values x=10 and y =1000 (I chose y = 1000 for simplicity, you'll know as I solve this)

50-16/100-10       *    100-10/100-10
34*90 / 90 * 90
=3060/8100
=0.377
Now case 2:

Code: [Select]

(5x-16)/ (√ y-10) * (√ y+10)/(√ y+10)

Plotting 10,1000

50-16/100-10 * 100+10/100+10
34/90   *     110*110
=0.377
Case 1 == Case 2, Proved. Theoretically it should work even after we solve the original
(5x-16) / (√y-10) either by  (√y-10) or (√y+10)

Let's find out
(5x-16) / (√y-10)

Case 1:

Code: [Select]

(5x-16) / (√y-10)   * (√y-10)/(√y-10)

(5x√y -50x -16√y + 160) / (√y-10)^2 
5x√y -50x-16√y+160 / y+100-20√y    -------equation x

For 10,1000 in equation x
5(10)(100) - 50(10) -10(100)+160/1000+100-20(100)

=3660/'-900

=-4.0666

Case 2 :

Code: [Select]

(5x-16)/ (√ y-10) * (√ y+10)/(√ y+10)
5x√y +50x -16√y -160 / y-100               ||Difference of square formula in
Plotting 10,1000
5(10)(100) + 50(10)-16(100)-160 /1000-100
5000+500-1600-160/900

=3740/900
=+4.155


Hence case 1 ! = case 2 (is not equal). What the fuck is going wrong here?!




[HTH]

as this is an error in thinking...

You are partially correct. Multiplying by sqrt(y) plus or minus ten gives you the same numerical answer. But thats not why you do it. You do it because rationalizing it makes the denominator easier to deal with. You can see im sure that y - 100 is infinitely easier to deal with than y +20sqrt(y) + 100.

That is why your cases in your first part equalled out... Alll you are doing is multiplying by one!

Though granted the sqrt of 1000 isnt 100.. Which is where the issue is in the second set of equations.

[PsychoRebellious]

That is why your cases in your first part equalled out... Alll you are doing is multiplying by one!

Though granted the sqrt of 1000 isnt 100.. Which is where the issue is in the second set of equations.

This is beyond embarrassing. Still I wonder if that is where the problem lies as the both equations are being divided and multiplied by the same number. That shouldn't be causing the problem, right?

[HTH]

The issue is in the denominator you set Y equal to 1000 and you also set the square root of why to be equal to 100 if you redo it and set y to 10,000 I'm sure the answers will come out correct

[PsychoRebellious]

Taking x as 2 and y as 16 doesn't work either.
Once case generates -105 and the other 0.17

[HTH]

I cant be sure what you did but it works for me, as well as... Well everyone.

Multiplying by 1, no matter the form. Always gives you the same number back.

[PsychoRebellious]

I cant be sure what you did but it works for me, as well as... Well everyone.

Multiplying by 1, no matter the form. Always gives you the same number back.

What values of x and y are you trying to plot? I only wonder if using two different formulas (a+b)^2 and a^2-b^2 is causing some problem

[mushinz]

The thing is that they just want you to rationalize the denominator...they don't care about the result


This is what you're doing:

Code: [Select]

(5x-16) / (√y-10)       * (√ y-10)/(√ y-10)  =


                                        here you use (a-b)^2=a^2-2*a*b+b^2


[(5x-16)(√y-10)]/[y-2*√y*10+100]


and as you see you still have root of y in the denominator


and if you use


(5x-16) / (√y-10)       * (√ y+10)/(√ y+10)


you get


[(5x-16)(√y-10)]/[y-100]


so there is no root in the denominator