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Question Number 1041 by tera last updated on 22/May/15

if the roots of the equation ax^2 +5x−2=0 is 2 and b. then 4a^2 −4ab+b^2 =...

$${if}\:{the}\:{roots}\:{of}\:{the}\:{equation}\:{ax}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{2}=\mathrm{0}\:{is}\:\mathrm{2}\:{and}\:{b}. \\ $$$$ \\ $$$${then}\:\mathrm{4}{a}^{\mathrm{2}} −\mathrm{4}{ab}+{b}^{\mathrm{2}} =... \\ $$

Answered by prakash jain last updated on 23/May/15

Since x=2 is a root (x−2) is a factor of ax^2 +5x−2 Put x=2 4a+10−2=0⇒a=−2 −2x^2 +5x−2=0 −2x^2 +4x+x−2=0 −(2x−1)(x−2)=0 Roots at 2 and (1/2), hence b=(1/2) 4a^2 −4ab+b^2 =4(−2)^2 −4(−2)((1/2))+((1/2))^2 =16+4+(1/4)=((81)/4)

$$\mathrm{Since}\:{x}=\mathrm{2}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\left({x}−\mathrm{2}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor}\:\mathrm{of} \\ $$$${ax}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{2} \\ $$$$\mathrm{Put}\:{x}=\mathrm{2} \\ $$$$\mathrm{4}{a}+\mathrm{10}−\mathrm{2}=\mathrm{0}\Rightarrow{a}=−\mathrm{2} \\ $$$$−\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{2}=\mathrm{0} \\ $$$$−\mathrm{2}{x}^{\mathrm{2}} +\mathrm{4}{x}+{x}−\mathrm{2}=\mathrm{0} \\ $$$$−\left(\mathrm{2}{x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)=\mathrm{0} \\ $$$$\mathrm{Roots}\:\mathrm{at}\:\mathrm{2}\:\mathrm{and}\:\frac{\mathrm{1}}{\mathrm{2}},\:\mathrm{hence}\:{b}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{4}{a}^{\mathrm{2}} −\mathrm{4}{ab}+{b}^{\mathrm{2}} =\mathrm{4}\left(−\mathrm{2}\right)^{\mathrm{2}} −\mathrm{4}\left(−\mathrm{2}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$=\mathrm{16}+\mathrm{4}+\frac{\mathrm{1}}{\mathrm{4}}=\frac{\mathrm{81}}{\mathrm{4}} \\ $$

Commented by tera last updated on 23/May/15

thank you sir

$${thank}\:{you}\:{sir} \\ $$

Answered by prakash jain last updated on 22/May/15

sum of roots 2+b=((−5)/a)⇒2a+ab=−5 product of roots 2b=((−2)/a)⇒ab=−1 2a+ab=−5⇒2a−1=−5⇒a=−2 ab=−1⇒b=(1/2) put values in 4a^2 −4ab+b^2

$${sum}\:{of}\:{roots}\:\mathrm{2}+{b}=\frac{−\mathrm{5}}{{a}}\Rightarrow\mathrm{2}{a}+{ab}=−\mathrm{5} \\ $$$${product}\:{of}\:{roots}\:\mathrm{2}{b}=\frac{−\mathrm{2}}{{a}}\Rightarrow{ab}=−\mathrm{1} \\ $$$$\mathrm{2}{a}+{ab}=−\mathrm{5}\Rightarrow\mathrm{2}{a}−\mathrm{1}=−\mathrm{5}\Rightarrow{a}=−\mathrm{2} \\ $$$${ab}=−\mathrm{1}\Rightarrow{b}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${put}\:{values}\:{in}\:\mathrm{4}{a}^{\mathrm{2}} −\mathrm{4}{ab}+{b}^{\mathrm{2}} \\ $$

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