Question

if a and ß are zeroes of quadratic polynomial 2x²+5x+2 then find the value of a⁴+ß⁴​

Answer 1

Step-by-step explanation:

Given :-

a and ß are zeroes of quadratic polynomial 2x²+5x+2 .

To find :-

Find the value of a⁴+ß⁴?

Solution:-

Given Quadratic Polynomial=P(x) = 2x²+5x+2

On Comparing this with the standard quadratic Polynomial ax²+bx+c

We have,

a = 2

b = 5

c = 2

Given zeroes = a and ß

We know that

Sum of the zeroes = -b/a

=> a+ß = -5/2 -------(1)

=> ß = (-5/2)-a ------(2)

and

Product of the zeroes = c/a

=> a ß = 2/2

=> aß = 1-------(3)

We know that

(a-b)² = (a+b)²-4ab

=> (a-ß)² = (a+ß)²-4(aß)

=> (a-ß)²= (-5/2)²-4(1)

=>(a-ß)²= (25/4)-4

=>(a-ß)²=(25-16)/4

=> (a-ß)² = 9/4

=>(a-ß) =3/2---(4) (taking positive value )

On adding (1)&(4) then

a+ß+a-ß = (-5/2)+(3/2)

=> 2a = (-5+3)/2

=> 2a = -2/2

=> 2a = -1

=> a = -1/2

On Substituting the value of a in (2) then

ß = (-5/2)-(-1/2)

=> ß = (-5/2)+(1/2)

=> ß = (-5+1)/2

=> ß = -4/2

=> ß = -2

We have , a =-1/2 ,ß = -2

The value of a⁴+ß⁴

=> (-1/2)⁴+(2)⁴

=> (1/16)+16

=>(1+16×16)/16

=>(1+256)/16

=>257/16

Answer:-

The value of a⁴+ß⁴ for the given problem is 257/16

Used formulae:-

• The standard quadratic polynomial is ax²+bx+c
• Sum of the zeroes = -b/a
• Product of the zeroes = c/a
• (a-b)² = (a+b)²-4ab