Question

if alpha,beta are roots of 2x²-3x-6=0 then equation whose roots are aplha²+2,beta²+2 is ??

Answer 1

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                             ____________________________

Given,

α and ß are the roots of 2x² - 3x - 6 = 0.

We have,

Coefficient of x² ( a ) = 2

Coefficient of x ( b ) = -3.

Constant term  ( c ) = -6.

We know the relationship between zeroes and coefficient of a quadratic polynomial.

⇒ Sum of zeroes = -b / a

⇒ α + ß = - ( -3 ) / 2

⇒ α + ß = 3 / 2  ------- ( 1 )

 Product of zeroes = c /a

⇒ αß = -6 / 2

∴ αß = -3.

Now , 

⇒ α² + ß² = ( α + ß )² - 2αß

⇒ α² + ß² = ( 3/2 )² - 2 × ( - 3 )

⇒ α² + ß² = ( 9/4 ) + 6

⇒ α² + ß² = ( 9 + 24 ) / 4

⇒ α² + ß² = 33/4.

The general form of a quadratic equation is :

⇒ x² - ( sum of zeroes ) x + Product of zeroes = 0

⇒  x² - ( α² + 2 + ß² + 2 )x + ( α² + 2 ) ( ß² + 2 ) = 0

⇒  x² - ( α² + ß² + 4 ) x+ ( α²ß² + 2α² + 2ß² + 4 ) = 0

⇒  x² - { ( 33/4 ) + 4 }x + ( αß )² + 2 ( α² + ß² ) + 4 = 0

⇒ x² - { ( 33 + 16 ) / 4 }x + ( -3 )² + 2 ( 33/4 ) + 4 = 0

⇒ x²  -  (49/4) x + 9 + ( 33 / 2 ) + 4 = 0

    4x² - 49x + 36 + 66 + 16
⇒ ----------------------------------- = 0
                      4

        4x² - 49x + 118
⇒ --------------------------------- = 0
                    4

⇒ 4x² - 49x + 118 = 0

The required quadratic equation is ( 4x² - 49x + 118 = 0 ).


Hope it helps !

Answer 2

Answer:

alpha^2 + 2 X beta^ 2 + 2 = 85/2

Step-by-step explanation:

alpha = - b/a ; beta = c/a

The given equation is 2x^2 - 3x - 6 = 0

a = 2, b = -3, c = -6

alpha = - (-3) / 2 = 3/2; beta = -6 / 2 = -3

alpha ^2  + 2 X beta^ 2 + 2 = ?

= (3/2)^2 + 2 X (-3)^2 + 2

= 9/4 + 18 + 2

= 9/4 + 20

= 170/4 = 85/2