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The expansion of $${\left({\mathtt{2}}{\mathtt{\,-\,}}{\mathtt{px}}\right)}^{{\mathtt{6}}}$$ in ascending powers of $${\mathtt{x}}$$, as far as the $${{\mathtt{x}}}^{{\mathtt{2}}}$$, is $${\mathtt{64}}{\mathtt{\,\small\textbf+\,}}{\mathtt{Ax}}{\mathtt{\,\small\textbf+\,}}{\mathtt{135}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}$$.

Given $$$$$${\mathtt{p}}$$ > $${\mathtt{0}}$$, find the value of $${\mathtt{p}}$$ and the valuse of $${\mathtt{A}}$$.

 Oct 5, 2014

Best Answer 

 #1
avatar+128079 
+5

Expanding  (2 - px)^6 to the first three terms and  setting it equal to 64 + Ax + 135x^2, we have:

2^6 - (6)*2^5*(px) + 15*(2^4)*(px)^2 = 64 + Ax + 135x^2

64 - 192(px) + 240(px)^2 = 64 + Ax + 135x^2   ...   subtract 64 from both sides

-192(px) + 240(px)^2 = Ax + 135x^2

So, equating coefficients, we have

-192p = A      and  240p^2 = 135

So, solving for p^2 using the second equation, we have that p^2 = 135/240 = 9/16

And taking the positive square root, p = 3/4

And substituting for p in the first to find A, we have

-192(3/4) = A = -144

So...p = 3/4 and A = -144

 

 Oct 5, 2014
 #1
avatar+128079 
+5
Best Answer

Expanding  (2 - px)^6 to the first three terms and  setting it equal to 64 + Ax + 135x^2, we have:

2^6 - (6)*2^5*(px) + 15*(2^4)*(px)^2 = 64 + Ax + 135x^2

64 - 192(px) + 240(px)^2 = 64 + Ax + 135x^2   ...   subtract 64 from both sides

-192(px) + 240(px)^2 = Ax + 135x^2

So, equating coefficients, we have

-192p = A      and  240p^2 = 135

So, solving for p^2 using the second equation, we have that p^2 = 135/240 = 9/16

And taking the positive square root, p = 3/4

And substituting for p in the first to find A, we have

-192(3/4) = A = -144

So...p = 3/4 and A = -144

 

CPhill Oct 5, 2014

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