Menu Close

discussion-back-with-mr-W-consider-this-equation-x-2-2x-3-3-x-27-0-does-the-equation-have-two-roots-or-three-roots-

Tinku Tara 0 Comments
Pin




Question Number 79647 by john santu last updated on 27/Jan/20
discussion back with mr W. consider this equation (x^2 −2x−3)(3^x −27)=0 does the equation have two roots or three roots?
$$\mathrm{discussion}\:\mathrm{back}\:\mathrm{with}\:\mathrm{mr}\:\mathrm{W}. \\ $$$$\mathrm{consider}\:\mathrm{this}\:\mathrm{equation}\: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}\right)\left(\mathrm{3}^{\mathrm{x}} −\mathrm{27}\right)=\mathrm{0} \\ $$$$\mathrm{does}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{have}\:\mathrm{two}\:\mathrm{roots} \\ $$$$\mathrm{or}\:\mathrm{three}\:\mathrm{roots}? \\ $$
Commented by naka3546 last updated on 27/Jan/20
x ∈ R or x ∈ C ?
$${x}\:\in\:\mathbb{R}\:\:{or}\:\:{x}\:\in\:\mathbb{C}\:? \\ $$
Commented by john santu last updated on 27/Jan/20
x∈R sir
$$\mathrm{x}\in\mathbb{R}\:\mathrm{sir} \\ $$
Answered by MJS last updated on 27/Jan/20
x^2 −2x−3=0∨3^x −27=0 (x=−1∨x=3)∨(x=3)≡(x=−1∨x=3) 2 roots we have different kinds of roots, in this case 1 single root x=−1 1 double root x=3
$${x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}=\mathrm{0}\vee\mathrm{3}^{{x}} −\mathrm{27}=\mathrm{0} \\ $$$$\left({x}=−\mathrm{1}\vee{x}=\mathrm{3}\right)\vee\left({x}=\mathrm{3}\right)\equiv\left({x}=−\mathrm{1}\vee{x}=\mathrm{3}\right) \\ $$$$\mathrm{2}\:\mathrm{roots} \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{different}\:\mathrm{kinds}\:\mathrm{of}\:\mathrm{roots},\:\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$$\mathrm{1}\:\mathrm{single}\:\mathrm{root}\:{x}=−\mathrm{1} \\ $$$$\mathrm{1}\:\mathrm{double}\:\mathrm{root}\:{x}=\mathrm{3} \\ $$
Commented by MJS last updated on 27/Jan/20
f′(x)=2(x−1)(3^x −27)+(x^2 −2x−3)3^x ln 3 f′(−1)=((320)/3)>0 ⇒ { ((f(x)<0; x<−1)),((f(x)>0; x>−1)) :} f′(3)=0 ⇒ f(x)>0; −1<x<∞ at x=3 there′s also a local minimum counting roots means to count the x−values at which y becomes 0 after counting we classify these roots y=(x−3)^(17) has only one root but it′s a seventeen−fold root
$${f}'\left({x}\right)=\mathrm{2}\left({x}−\mathrm{1}\right)\left(\mathrm{3}^{{x}} −\mathrm{27}\right)+\left({x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}\right)\mathrm{3}^{{x}} \mathrm{ln}\:\mathrm{3} \\ $$$${f}'\left(−\mathrm{1}\right)=\frac{\mathrm{320}}{\mathrm{3}}>\mathrm{0}\:\Rightarrow\:\begin{cases}{{f}\left({x}\right)<\mathrm{0};\:{x}<−\mathrm{1}}\\{{f}\left({x}\right)>\mathrm{0};\:{x}>−\mathrm{1}}\end{cases} \\ $$$${f}'\left(\mathrm{3}\right)=\mathrm{0}\:\Rightarrow\:{f}\left({x}\right)>\mathrm{0};\:−\mathrm{1}<{x}<\infty \\ $$$$\mathrm{at}\:{x}=\mathrm{3}\:\mathrm{there}'\mathrm{s}\:\mathrm{also}\:\mathrm{a}\:\mathrm{local}\:\mathrm{minimum} \\ $$$$ \\ $$$$\mathrm{counting}\:\mathrm{roots}\:\mathrm{means}\:\mathrm{to}\:\mathrm{count}\:\mathrm{the}\:{x}−\mathrm{values} \\ $$$$\mathrm{at}\:\mathrm{which}\:{y}\:\mathrm{becomes}\:\mathrm{0} \\ $$$$\mathrm{after}\:\mathrm{counting}\:\mathrm{we}\:\mathrm{classify}\:\mathrm{these}\:\mathrm{roots} \\ $$$${y}=\left({x}−\mathrm{3}\right)^{\mathrm{17}} \:\mathrm{has}\:\mathrm{only}\:\mathrm{one}\:\mathrm{root}\:\mathrm{but}\:\mathrm{it}'\mathrm{s}\:\mathrm{a} \\ $$$$\mathrm{seventeen}−\mathrm{fold}\:\mathrm{root} \\ $$
Commented by john santu last updated on 27/Jan/20
ok. thank you mister
$$\mathrm{ok}.\:\mathrm{thank}\:\mathrm{you}\:\mathrm{mister} \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *