If alpha and beta are the zeros of the Quarditic polynomial p(x) = x2-x-4 then find 1/ alpha + 1/beta - alpha beta
Answers
Sol : We have quadratic equation x² - x - 4.
Given α and ß are their zeroes.
We know that,
Sum of roots = - ( coefficient of x )/ coefficient of x²
α + ß = - ( - 1 ) / 1
α + ß = 1 / 1 = 1.
Now,
Product of roots = constant term / coefficient of x²
αß = ( - 4 ) / 1
αß = -4.
1. 1/α + 1/ß - αß
ß + α
= ------------- - αß
αß
By substituting the values of ( α + ß ) and ( αß ),
= ( 1 / -4 ) - ( - 4 )
= ( - 1 / 4 ) + 4
- 1 + 16
= ----------------
4
= 15 / 4.
2. α/ß + ß/α + 2 ( 1/α + 1/ß ) + 3αß
α² + ß² + 2ß + 2α
= --------------------------- + 3αß
αß
α² + ß² + 2 ( α+ß )
= ------------------------- + 3αß ----- eq.1
αß
Now ,we don't have the value of ( α² + ß² ), so let's find it ,
( α + ß )² = α² + ß² + 2 αß
By substituting the values of ( α + ß ) and αß in above equation,
( 1 )² = α² + ß² + 2 ( - 4 )
1 = α² + ß² - 8
α² + ß² = 1 + 8
α² + ß² = 9
Now by substituting the values of ( α² + ß² ) ,αß and ( α + ß ) in eq.1,
9 + 2 ( 1 )
= --------------- + 3 ( - 4 )
-4
9 + 2
= -------------- - 12
-4
- 11
= -------------- - 12
4
-11 - 48
= --------------
4
= -59/4.
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Answer:
Sol : We have quadratic equation x² - x - 4.
Given α and ß are their zeroes
We know that,
Sum of roots = - ( coefficient of x )/ coefficient of x²
α + ß = - ( - 1 ) / 1
α + ß = 1 / 1 = 1.
Now
Product of roots = constant term / coefficient of x²
αß = ( - 4 ) / 1
αß = -4.
1. 1/α + 1/ß - αß
ß + α
= ------------- - αß
By substituting the values of ( α + ß ) and ( αß ),
= ( 1 / -4 ) - ( - 4 )
= ( - 1 / 4 ) + 4
- 1 + 16
= ----------------
4
= 15 / 4.
2. α/ß + ß/α + 2 ( 1/α + 1/ß ) + 3αß
α² + ß² + 2ß + 2α
= --------------------------- + 3αß
αß
α² + ß² + 2 ( α+ß )
= ------------------------- + 3αß ----- eq.1
αß
Now ,we don't have the value of ( α² + ß² ), so let's find it ,
( α + ß )² = α² + ß² + 2 αß
By substituting the values of ( α + ß ) and αß in above equation,
( 1 )² = α² + ß² + 2 ( - 4 )
1 = α² + ß² - 8
α² + ß² = 1 + 8
α² + ß² = 9
Now by substituting the values of ( α² + ß² ) ,αß and ( α + ß ) in eq.1,
9 + 2 ( 1 )
= --------------- + 3 ( - 4 )
-4
9 + 2
= -------------- - 12
-4
- 11 -48
4
=-59/4