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Question Number 115854 by bemath last updated on 29/Sep/20

If f(2x)= x^2 +4x+1 , what all values of t for which f((t/2)) = −((11)/4) where f represents a function

$${If}\:{f}\left(\mathrm{2}{x}\right)=\:{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{1}\:,\:{what}\:{all} \\ $$$${values}\:{of}\:{t}\:{for}\:{which}\:{f}\left(\frac{{t}}{\mathrm{2}}\right)\:=\:−\frac{\mathrm{11}}{\mathrm{4}} \\ $$$${where}\:{f}\:{represents}\:{a}\:{function} \\ $$

Answered by bobhans last updated on 29/Sep/20

⇔ f(x)= ((x/2))^2 +4((x/2))+1 ⇔ f((t/2)) = ((t/4))^2 +4((t/4))+1 = −((11)/4) ⇒ (t^2 /(16)) + ((4t)/4) + ((15)/4) = 0 ⇒ t^2 +16t +60=0 ⇒ (t+6)(t+10)=0 → { ((t=−6)),((t=−10)) :}

$$\:\Leftrightarrow\:{f}\left({x}\right)=\:\left(\frac{{x}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{4}\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{1} \\ $$$$\Leftrightarrow\:{f}\left(\frac{{t}}{\mathrm{2}}\right)\:=\:\left(\frac{{t}}{\mathrm{4}}\right)^{\mathrm{2}} +\mathrm{4}\left(\frac{{t}}{\mathrm{4}}\right)+\mathrm{1}\:=\:−\frac{\mathrm{11}}{\mathrm{4}} \\ $$$$\Rightarrow\:\frac{{t}^{\mathrm{2}} }{\mathrm{16}}\:+\:\frac{\mathrm{4}{t}}{\mathrm{4}}\:+\:\frac{\mathrm{15}}{\mathrm{4}}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:{t}^{\mathrm{2}} +\mathrm{16}{t}\:+\mathrm{60}=\mathrm{0} \\ $$$$\Rightarrow\:\left({t}+\mathrm{6}\right)\left({t}+\mathrm{10}\right)=\mathrm{0}\:\rightarrow\begin{cases}{{t}=−\mathrm{6}}\\{{t}=−\mathrm{10}}\end{cases} \\ $$

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