Question Number 101056 by bemath last updated on 30/Jun/20

If the equation 4x^2 −4(5x+1)+p^2 =0  has one root equals to two more  then the other, then the value of  p is equal to ___

Commented bybemath last updated on 30/Jun/20

thank you both

Answered by Rasheed.Sindhi last updated on 30/Jun/20

Still another way  4x^2 −20x+p^2 −4=0.......(i)  The equation with α & α+2 roots     (x−α)(x−α−2)=0     x^2 −(2α+2)x+α^2 +2α=0....(ii)  Comparing coefficients of (i)&(ii)  (4/1)=((−20)/(−(2α+2)))=((p^2 −4)/(α^2 +2α))  8α+8=20 ∧ p^2 −4=4α^2 +8α  α=(3/2)  ∧ p^2 =4((3/2))^2 +8((3/2))+4         p^2 =9+12+4=25                   p=±5

Commented bybemath last updated on 30/Jun/20

waw...great sir. thank you

Answered by Rasheed.Sindhi last updated on 30/Jun/20

4x^2 −4(5x+1)+p^2 =0  4x^2 −20x−4+p^2 =0  α+(α+2)=((−(−20))/4) ∧ α(α+2)=((p^2 −4)/4)  2α+2=5⇒α=(3/2)      ⇒((p^2 −4)/4)=α(α+2)=(3/2)((3/2)+2)      ⇒((p^2 −4)/4)=(3/2)((7/2))  ⇒p^2 =21+4⇒p=±5

Answered by Ar Brandon last updated on 30/Jun/20

Let the roots be α and α+2. Then;  Sum of roots 2α+2=((20)/4)=5⇒α=(3/2)  Product of roots α(α+2)=α^2 +2α=((p^2 −4)/4)  ⇒((3/2))^2 +2((3/2))=((p^2 −4)/4)=((21)/4)⇒p=±5

Answered by Rasheed.Sindhi last updated on 30/Jun/20

AnOther Way  4x^2 −20x+p^2 −4=0  α is a root (say)  4α^2 −20α+p^2 −4=0.......(i)  α+2 is other root  4(α+2)^2 −20(α+2)+p^2 −4=0  4α^2 +16α+16−20α−40+p^2 −4=0  4α^2 −4α+p^2 −28=0.......(ii)  (i)−(ii):−16α+24=0⇒α=(3/2)  Now ,  (i)⇒4((3/2))^2 −20((3/2))+p^2 −4=0              9−30+p^2 −4=0⇒p=±5

Answered by Rasheed.Sindhi last updated on 30/Jun/20

One way more:A simple way  (Whether you like it or dislike,      anyway this is also a way.)   4x^2 −20x+p^2 −4=0  x=((5±(√(29−p^2 )))/2)  ((5+(√(29−p^2 )))/2)−((5−(√(29−p^2 )))/2)=2  ((5+(√(29−p^2 ))−5+(√(29−p^2 )))/2)=2  (√(29−p^2 ))=2  29−p^2 =4  p^2 =25  p=±5

Commented byjohn santu last updated on 01/Jul/20

cooll

Commented byRasheed.Sindhi last updated on 01/Jul/20

Thanks Sir!