# How do I manipulate this equation : $A(x^2+x)-B(x^2-x) = 4x^2-10x$ so i can solve for A and B?

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!

• 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... – Matti P. Oct 23 '18 at 11:29
• Yea that is my question @MattiP. excuse me sir can you please help me out – Nhoj_Gonk Oct 23 '18 at 11:30
• 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. – user376343 Oct 23 '18 at 11:36
• First time someone referred to me as a mathematician. Thank you. – Mohammad Zuhair Khan Oct 23 '18 at 12:47

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

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.

• 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! – Nhoj_Gonk Oct 23 '18 at 11:31
• edited, have a look – pooja somani Oct 23 '18 at 11:32
• can stack exchange give this guy a medal?? Thank you so much from Australia!<3 – Nhoj_Gonk Oct 23 '18 at 11:33
• you got it clear now? – pooja somani Oct 23 '18 at 11:34
• yes sir! <3 Thank you again! – Nhoj_Gonk Oct 23 '18 at 11:35

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.

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?

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