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If-f-x-x-2-ax-b-if-f-1-3-and-and-one-of-the-roots-of-the-eqiation-f-x-0-doubles-the-other-find-the-positive-values-of-a-and-b-




Question Number 210013 by otchereabdullai@gmail.com last updated on 28/Jul/24
 If f(x)= x^2 +ax+b. if f(1)= 3 and    and one of the roots of the eqiation   f(x)= 0 doubles the other, find the   positive values of a and b.
$$\:{If}\:{f}\left({x}\right)=\:{x}^{\mathrm{2}} +{ax}+{b}.\:{if}\:{f}\left(\mathrm{1}\right)=\:\mathrm{3}\:{and}\: \\ $$$$\:{and}\:{one}\:{of}\:{the}\:{roots}\:{of}\:{the}\:{eqiation} \\ $$$$\:{f}\left({x}\right)=\:\mathrm{0}\:{doubles}\:{the}\:{other},\:{find}\:{the} \\ $$$$\:{positive}\:{values}\:{of}\:{a}\:{and}\:{b}. \\ $$
Answered by mr W last updated on 28/Jul/24
say the one root is p, then the other  root is 2p.  p+2p=−a ⇒p=−(a/3)  p×2p=b ⇒2a^2 =9b  f(1)=1^2 +a+b=3 ⇒a+b=2  2a^2 =9(2−a)  ⇒2a^2 +9a−18=0  ⇒a= (3/2), −6 (rejected)  ⇒b=(1/2)
$${say}\:{the}\:{one}\:{root}\:{is}\:{p},\:{then}\:{the}\:{other} \\ $$$${root}\:{is}\:\mathrm{2}{p}. \\ $$$${p}+\mathrm{2}{p}=−{a}\:\Rightarrow{p}=−\frac{{a}}{\mathrm{3}} \\ $$$${p}×\mathrm{2}{p}={b}\:\Rightarrow\mathrm{2}{a}^{\mathrm{2}} =\mathrm{9}{b} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1}^{\mathrm{2}} +{a}+{b}=\mathrm{3}\:\Rightarrow{a}+{b}=\mathrm{2} \\ $$$$\mathrm{2}{a}^{\mathrm{2}} =\mathrm{9}\left(\mathrm{2}−{a}\right) \\ $$$$\Rightarrow\mathrm{2}{a}^{\mathrm{2}} +\mathrm{9}{a}−\mathrm{18}=\mathrm{0} \\ $$$$\Rightarrow{a}=\:\frac{\mathrm{3}}{\mathrm{2}},\:−\mathrm{6}\:\left({rejected}\right) \\ $$$$\Rightarrow{b}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by otchereabdullai@gmail.com last updated on 28/Jul/24
thanks soo much Prof W
$${thanks}\:{soo}\:{much}\:{Prof}\:{W} \\ $$

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