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I was given this polynomial equation $$A(x^2+x)-B(x^2-x) = 4x^2-10x$$ and I was given a clue in order to solve A and B, it is best to convert from the original equation to this :$$(A-B)x^2+(A+B)x = 4x^2-10x.$$

In general, can you show me how to convert $A(x^2+x)-B(x^2-x) = 4x^2-10x$ to $(A-B)x^2+(A+B)x = 4x^2-10x$?

Are there some kind of manipulations that I need to know?

Thank you every much!

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    $\begingroup$ So you're asking how to convert $$ A(x^2 + x) - B(x^2 -x ) = 4x^2 -10x $$ into $$ (A-B)x^2 + (A+B)x = 4x^2 -10x $$ ? Answer: By opening the parentheses and juggling the terms around, it's not particularly difficult... $\endgroup$ – Matti P. Oct 23 '18 at 11:29
  • $\begingroup$ Yea that is my question @MattiP. excuse me sir can you please help me out $\endgroup$ – Nhoj_Gonk Oct 23 '18 at 11:30
  • $\begingroup$ Opening parenthesis gives $Ax^2+Ax-Bx^2-(-1)Bx=4x^2-10x.$ Now put together terms with $x^2$ and together the terms with $x.$ This gives what @Matti P wrote. $\endgroup$ – user376343 Oct 23 '18 at 11:36
  • $\begingroup$ First time someone referred to me as a mathematician. Thank you. $\endgroup$ – Mohammad Zuhair Khan Oct 23 '18 at 12:47
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$$A(x^2+x)-B(x^2-x) = 4x^2-10x$$

Open the brackets. It can be written as: $$Ax^2+Ax-B x^2-B(-x)=4x^2-10x$$ Or,$$Ax^2+Ax-B x^2+Bx=4x^2-10x.$$ Take $x^2$ common from $Ax^2$ and $Bx^2$ and $x$ common from $Ax$ and $Bx.$

$$(A-B)x^2+(A+B)x=4x^2-10x$$

Now compare both sides:

Equate terms having $x^2$ and those having x. you get:$$A-B=4$$ AND $$A+B=-10.$$ Solve simultaneously.

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  • $\begingroup$ yes i have already got this answer since i was given the clue, but how do i algebraically convert this $A(x^2+x)-B(x^2-x) = 4x^2-10x$ to $(A-B)x^2+(A+B)x = 4x^2-10x$ and also thank you for answering! $\endgroup$ – Nhoj_Gonk Oct 23 '18 at 11:31
  • $\begingroup$ edited, have a look $\endgroup$ – pooja somani Oct 23 '18 at 11:32
  • $\begingroup$ can stack exchange give this guy a medal?? Thank you so much from Australia!<3 $\endgroup$ – Nhoj_Gonk Oct 23 '18 at 11:33
  • $\begingroup$ you got it clear now? $\endgroup$ – pooja somani Oct 23 '18 at 11:34
  • $\begingroup$ yes sir! <3 Thank you again! $\endgroup$ – Nhoj_Gonk Oct 23 '18 at 11:35
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Take common factor $x^2$ and $x$ from both expressions. You'll be left with a polynomial of order two, and then you can solve for $A$ and $B$ by equating the resulting coefficients to $4$ and $-10$ respectively.

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So we are staring with $$ A(x^2 + x) - B(x^2 -x ) $$ Open the brackets, and the expressions becomes $$ Ax^2 + Ax - Bx^2 + Bx $$ For the context of the problem, we want to group the $x$'s and $x^2$ together, like so: $$ Ax^2 - Bx^2 + Ax + Bx $$ And now factoring the terms together, we get $$ (A-B)x^2 + (A+B) x $$ Any questions?

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  • $\begingroup$ No more sir! This explained it all! Thank you prof Matti from Australia! $\endgroup$ – Nhoj_Gonk Oct 23 '18 at 11:34

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