Question

sum and product of zeros of a quadratic polynomial are 2 and √23 respectively then find that polynomial.​

Answer 1

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▪Given :-

For a Quadratic Polynomial

   

Sum of Zeros = 2

Product of Zeros = √23

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▪To Find :-

The Quadratic Polynomial.

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▪Key Concept :-

If sum and product of zeros of any quadratic polynomial are s and p respectively,

Then,

The quadratic polynomial is given by :-

$\bf  {x}^{2}  - s \: x + p$

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▪Solution :-

Here,

Sum = s = 2

and

Product = p = √23

So,

Required Polynomial should be

$\bf{x}^{2}  - 2x +  \sqrt{23}$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

Answer:

Given :-

• Sum and product of zeroes of a quadratic polynomial are 2 and √23 respectively.

To Find :-

• What is the polynomial.

Formula Used :-

$\clubsuit$ Quadratic Equation Formula :

$\footnotesize\mapsto \sf\boxed{\bold{\pink{x^2 - (Sum\: of\: roots)x + (Product\: of\: roots) =\: 0}}}$

Solution :-

Given :

$\bigstar \: \: \sf\bold{\purple{Sum\: of\: roots\: (\alpha + \beta) =\: 2}}$

$\bigstar\: \: \: \sf\bold{\purple{Product\: of\: roots\: (\alpha\beta) =\: \sqrt{23}}}$

Hence, the required polynomial equation is :

$\footnotesize\leadsto \sf\bold{\green{x^2 - (Sum\: of\: roots)x + (Product\: of\: roots) =\: 0}}$

$\small\longrightarrow \bf{ x^2 - (\alpha + \beta)x + (\alpha\beta) =\: 0}$

$\small\longrightarrow \sf x^2 - (2)x + (\sqrt{23}) =\: 0$

$\small\longrightarrow \sf x^2 - 2x + \sqrt{23} =\: 0$

$\small\longrightarrow \sf\bold{\red{x^2 - 2x + \sqrt{23} =\: 0}}$

${\small{\bold{\underline{\therefore\: The\: required\: polynomial\: equation\: is\: x^2 - 2x + \sqrt{23}\: =\: 0\: .}}}}$

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EXTRA INFORMATION :-

❒ Quadratic Equation with one variable :

✪ The general form of the equation is ax² + bx + c.

[ Note :- ● If a = 0, then the equation becomes to a linear equation. ]

● If b = 0, then the roots of the equation becomes equal but opposite in sign. ]

● If c = 0, then one of the roots is zero. ]

❒ Nature of Roots :

✪ b² - 4ac is the discriminant of the equation.

Then,

◆ If, b² - 4ac = 0, then the roots are real & equal.

◆ If, b² - 4ac > 0, then the roots are real & unequal

◆ If, b² - 4ac < 0, then the roots are imaginary & no real roots.