Question

Find the quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively

1/4, -1​

Answer 1

Sum of the zeros
= -b/a
= 1/4

Product of the zeros
= c/a
= -1/1
= (-1×4) / (1×4)
= -4/4

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

Hence the quadratic polynomial is
4x^2 - x - 4 = 0

Answer 2

Aɴꜱᴡᴇʀ

$\huge \sf \pink{4 {x}^{2} - x - 4}$

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Gɪᴠᴇɴ

Sum and the product of its zeros are $\frac{1}{4}$ and -1

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ᴛᴏ ꜰɪɴᴅ

The quadratic polynomial

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Sᴛᴇᴘꜱ

$\begin{lgathered}\implies\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:(x)^{2} }{Coefficient\:of\:(x)} }\\\\\\\mapsto\sf{\dfrac{1}{4} =\dfrac{-b}{a} }\\\\\\\mapsto\sf{1 \times a=-4b}\\\\\\\mapsto\sf{1=-4b}\\\\\\\mapsto\sf{\red{b=-\dfrac{1}{4} }}\end{lgathered}$

$\mathbb{\pink{\boxed{\fbox{\bf{Product\:of\:the\:zeroes\::}}}}}$

$\begin{lgathered}\implies\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term }{Coefficient\:of\:(x)} }\\\\\\\mapsto\sf{-1 =\dfrac{c}{a} }\\\\\\\mapsto\sf{-1 \times a=c}\\\\\\\mapsto\sf{-1=c}\\\\\\\mapsto\sf{\red{c=-1 }}\end{lgathered}$

We have;

a = 1

b = -1/4

c = -1

We know that formula of the quadratic equation :

$\begin{lgathered}\implies\sf{x^{2} -(sum\:of\:zeroes)+(product\:of\:zeroes)}\\\\\ = \sf{x^{2} -\dfrac{1}{4} x+(-1)}\\\\\leadsto\sf{\red{4x^{2} -x-4}}\end{lgathered}$

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$\huge{\mathfrak{\purple{hope\; it \;helps}}}$