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Question Number 177856 by mathlove last updated on 10/Oct/22

p(x)+p(x−1)=2x+4  p(x)=?

$${p}\left({x}\right)+{p}\left({x}−\mathrm{1}\right)=\mathrm{2}{x}+\mathrm{4} \\ $$$${p}\left({x}\right)=? \\ $$

Answered by blackmamba last updated on 10/Oct/22

  p(x)=ax+b     p(x−1)=a(x−1)+b=ax+b−a  ⇒p(x)+p(x−1)=2ax+2b−a=2x+4     { ((2a=2⇒a=1)),((2b−2=4⇒b=3)) :}   ∴ p(x)=2x+3

$$\:\:{p}\left({x}\right)={ax}+{b}\: \\ $$$$\:\:{p}\left({x}−\mathrm{1}\right)={a}\left({x}−\mathrm{1}\right)+{b}={ax}+{b}−{a} \\ $$$$\Rightarrow{p}\left({x}\right)+{p}\left({x}−\mathrm{1}\right)=\mathrm{2}{ax}+\mathrm{2}{b}−{a}=\mathrm{2}{x}+\mathrm{4} \\ $$$$\:\:\begin{cases}{\mathrm{2}{a}=\mathrm{2}\Rightarrow{a}=\mathrm{1}}\\{\mathrm{2}{b}−\mathrm{2}=\mathrm{4}\Rightarrow{b}=\mathrm{3}}\end{cases} \\ $$$$\:\therefore\:{p}\left({x}\right)=\mathrm{2}{x}+\mathrm{3} \\ $$

Commented by mathlove last updated on 10/Oct/22

thanks

$${thanks} \\ $$

Commented by Tinku Tara last updated on 10/Oct/22

Small caculation error. pls check  a=1  2b−a=4⇒b=2.5  p(x)=2x+2.5

$$\mathrm{Small}\:\mathrm{caculation}\:\mathrm{error}.\:\mathrm{pls}\:\mathrm{check} \\ $$$${a}=\mathrm{1} \\ $$$$\mathrm{2}{b}−{a}=\mathrm{4}\Rightarrow{b}=\mathrm{2}.\mathrm{5} \\ $$$${p}\left({x}\right)=\mathrm{2}{x}+\mathrm{2}.\mathrm{5} \\ $$

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