Question

To find the quadratic polynomial each with the given number as the sum and product of its zeros respectively 1/4,-1

Answer 1

Answer:4x²-x-4

Step-by-step explanation:

         α+β=1/4

         αβ= -1

required polynomial=k(x²-(α+β)x+αβ

                                =k(x²-1/4 -1 )

                                =k/4(4x²-x -4)   =required polynomial is 4x²-x-4

Answer 2

Given that,

Sum of Quadratic polynomial = ¼

& Product of Quadratic polynomial = – 1.

⠀

$\underline{\bigstar\:\textsf{Required Quadratic Polynomial \: :}} \\ \\ \\ :\implies\sf p(x) = x^2 - (\alpha + \beta)x + \alpha \: \beta \\\\\\:\implies\sf p(x) = x^2 - \dfrac{1}{4}x + (- 1) \\\\\\:\implies\sf p(x) = x^2 - \dfrac{1}{4}x - 1 \\\\\\:\implies\underline{\boxed{\frak{\purple{\pmb{p(x) = 4x^2 - x -4}}}}}\;\bigstar$

⠀

$\therefore{\underline{\sf{Hence,~the~Quadratic~polynomial~is~ \sf {\pmb{4x^2 - x - 4}.}}}}$

$\rule{100px}{.3ex}$

⠀

An Quadratic equation can be represent as in the form of (ax² + bx + c = 0).

If α and β are roots of any Quadratic equation (ax² + bx + c = 0) then Sum and Product is given by :

⠀⠀⠀⠀⋆ Sum (α + β) = (–b)/a

⠀⠀⠀⠀⋆ Product (α β) = c/a