Question

If a and b are zeroes of 4x^2-x-4 find quadratic polynomial whose zeroes are 1/2a and 1/2b​

Answer 1

Step-by-step explanation:

Hi friend !

p(x) = 4x² - x - 4

a = 4 , b = -1 , c = -4

α and β are zeros of p(x)

we know that ,

sum of zeros = -b/a

that is ,

α + β = -b/a = 1/4

product of zeros = c/a

that is ,

αβ = -4/4 = -1

==========================

1/2α and 1/2β are zeros of a polynomial

sum of zeros = 1/2α + 1/2β

= 2α + 2β / 4αβ

= 2 [α + β] / 4αβ

= [2 × 1/4] / 4× - 1

= (1/2)/ -4

= -1/8

product of zeros =( 1/2α )( 1/2β)

= 1/4αβ

= 1/4×-1

= -1/4

a quadratic polynomial is given by ,

k {x² - [ sum of zeros ]x + [ product of zeros ]}

Answer 2

Answer:

to solve this lets divide the quadratic equation by 4 such that we have it in the form x

2 +(α+β)x+ab

so after dividing by 4 we have

4x² +4x+1

_________

4

x² +x+ 1

___

4

So we have α+β=1

and αb = 1

___

4

So if zeroes to the new quadratic equation are 2α and 2β, then

2α+2β=2(α+β)

=2(1)=2

and,

2α×2β=4αβ

=4× 1

___= 1

4

So the new quadratic equation with its roots 2αand2β will be

x² +(2α+2β)x+2α×2β

Putting the values, the equation would be,

x² +2x+1

Hope it helps you