Menu Close

Given-that-k-2-3k-5-0-determine-the-value-of-k-4-6k-3-9k-2-7-




Question Number 198902 by necx122 last updated on 25/Oct/23
Given that k^2 −3k+5=0, determine  the value of k^4 −6k^3 +9k^2 −7
$${Given}\:{that}\:{k}^{\mathrm{2}} −\mathrm{3}{k}+\mathrm{5}=\mathrm{0},\:{determine} \\ $$$${the}\:{value}\:{of}\:{k}^{\mathrm{4}} −\mathrm{6}{k}^{\mathrm{3}} +\mathrm{9}{k}^{\mathrm{2}} −\mathrm{7} \\ $$
Answered by witcher3 last updated on 25/Oct/23
(k^2 −3k+5)(k^2 −3k−5)+18=k^4 −6k^3 +9k^2 −7  =0.(k^2 −3k−5)+18=18
$$\left(\mathrm{k}^{\mathrm{2}} −\mathrm{3k}+\mathrm{5}\right)\left(\mathrm{k}^{\mathrm{2}} −\mathrm{3k}−\mathrm{5}\right)+\mathrm{18}=\mathrm{k}^{\mathrm{4}} −\mathrm{6k}^{\mathrm{3}} +\mathrm{9k}^{\mathrm{2}} −\mathrm{7} \\ $$$$=\mathrm{0}.\left(\mathrm{k}^{\mathrm{2}} −\mathrm{3k}−\mathrm{5}\right)+\mathrm{18}=\mathrm{18} \\ $$
Commented by necx122 last updated on 25/Oct/23
Thank you so much sir
Commented by witcher3 last updated on 25/Oct/23
y′re welcom
$$\mathrm{y}'\mathrm{re}\:\mathrm{welcom} \\ $$
Answered by Rasheed.Sindhi last updated on 25/Oct/23
k^2 −3k+5=0;k^4 −6k^3 +9k^2 −7=?  ⇒k^2 =3k−5  ⇒k^3 =3k^2 −5k=3(3k−5)−5k=4k−15  ⇒k^4 =4k^2 −15k=4(3k−5)−15k=−3k−20  k^4 −6k^3 +9k^2 −7  =(−3k−20)−6(4k−15)+9(3k−5)−7  =−3k−20−24k+90+27k−45−7  =70−52=18
$${k}^{\mathrm{2}} −\mathrm{3}{k}+\mathrm{5}=\mathrm{0};{k}^{\mathrm{4}} −\mathrm{6}{k}^{\mathrm{3}} +\mathrm{9}{k}^{\mathrm{2}} −\mathrm{7}=? \\ $$$$\Rightarrow{k}^{\mathrm{2}} =\mathrm{3}{k}−\mathrm{5} \\ $$$$\Rightarrow{k}^{\mathrm{3}} =\mathrm{3}{k}^{\mathrm{2}} −\mathrm{5}{k}=\mathrm{3}\left(\mathrm{3}{k}−\mathrm{5}\right)−\mathrm{5}{k}=\mathrm{4}{k}−\mathrm{15} \\ $$$$\Rightarrow{k}^{\mathrm{4}} =\mathrm{4}{k}^{\mathrm{2}} −\mathrm{15}{k}=\mathrm{4}\left(\mathrm{3}{k}−\mathrm{5}\right)−\mathrm{15}{k}=−\mathrm{3}{k}−\mathrm{20} \\ $$$${k}^{\mathrm{4}} −\mathrm{6}{k}^{\mathrm{3}} +\mathrm{9}{k}^{\mathrm{2}} −\mathrm{7} \\ $$$$=\left(−\mathrm{3}{k}−\mathrm{20}\right)−\mathrm{6}\left(\mathrm{4}{k}−\mathrm{15}\right)+\mathrm{9}\left(\mathrm{3}{k}−\mathrm{5}\right)−\mathrm{7} \\ $$$$=−\mathrm{3}{k}−\mathrm{20}−\mathrm{24}{k}+\mathrm{90}+\mathrm{27}{k}−\mathrm{45}−\mathrm{7} \\ $$$$=\mathrm{70}−\mathrm{52}=\mathrm{18} \\ $$

Leave a Reply

Your email address will not be published.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