Question

Find the quadratic polynomial with given numbers sum of numbers=1/4, product of numbers= -1.​

Answer 1

Answer:

Given,

Sum of the roots of a Quadratic Polynomial = $\sf\dfrac{1}{4}$

Product of the roots of a Quadratic Polynomial = -1

To Find :-

The Quadratic Polynomial.

Solution :-

Sum of the roots of a Quadratic Polynomial = $\sf\dfrac{1}{4}$

Product of the roots of a Quadratic Polynomial = -1

Let, roots of the Quadratic Polynomial be :-

$\sf\alpha , \beta$

$\sf\implies Sum\: of \: the \: roots = \alpha + \beta = \dfrac{1}{4}$

Product of the roots = $\sf\alpha\times\beta = -1$

Formula Required :-

Quadratic Polynomial with roots :-

$\sf x^{2}- (\alpha + \beta)x+ (\alpha\times \beta) = 0$

Substituting value :-

$\sf x^{2}- \big(\dfrac{1}{4}\big)x+ (-1) = 0$

$\sf x^{2}- \big(\dfrac{x}{4}\big) - 1 = 0$

Taking L.C.M as 4 :-

$\sf (x^{2}\times \dfrac{4}{4})- \dfrac{x}{4}- (1\times \dfrac{4}{4}) = 0$

$\sf\dfrac{4x^{2}}{4}- \dfrac{x}{4}- \dfrac{4}{4} = 0$

$\sf\dfrac{4x^{2} - x - 4}{4}$ = 0

$\sf 4x^{2} - x - 4 = 0\times 4$

$\sf 4x^{2} - x - 4 = 0$

Answer 2

$\huge\red{\underline{\mathfrak{Solution}}}$

sum of zeroes is α+ β = 1/4

product of zeroes is αβ= -1

The formula for making quadratic polynomial

$K( {x}^{2} - ( \alpha + \beta )x + \alpha \beta )$

$K( {x}^{2} - \frac{1}{4} + ( - 1))$

$\frac{K}{4} ( {4x}^{2} - x - 4)$

The required polynomial is

$\boxed{\pink{\sf{4x²-x-4}}}$