Question

Find the quadratic polynomial , the sum of whose roots is √2 and their product is 1/4

Answer 1

Answer:

Given :-

• The sum of whose roots is √2 and their products is 1/4.

To Find :-

• What is the quadratic 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 :

$\bigstar\: \bf{Sum\: of\: roots\: (\alpha + \beta) =\: \sqrt{2}}$

$\bigstar\: \bf{Product\: of\: roots\: (\alpha\beta) =\: \dfrac{1}{4}}$

According to the question by using the formula we get,

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

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

$\small\longrightarrow \sf x^2 - (\sqrt{2})x + \dfrac{1}{4} =\: 0$

$\small\longrightarrow \sf x^2 - \sqrt{2}x + \dfrac{1}{4} =\: 0$

$\small\longrightarrow \sf \dfrac{4x^2 - 4\sqrt{2}x + 1}{4} =\: 0$

By doing cross multiplication we get,

$\small\longrightarrow \sf 4x^2 - 4\sqrt{2}x + 1 =\: 0(4)$

$\small\longrightarrow \sf\bold{\red{4x^2 - 4\sqrt{2}x + 1 =\: 0}}$

${\small{\bold{\underline{\therefore\: The\: required\: quadratic\:polynomial\: is\: 4x^2 - 4\sqrt{2}x + 1\: .}}}}$

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

❒ Nature of Roots :

✪ The discriminant of the equation is b² - 4ac. Then,

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

◆ b² - 4ac > 0, then the roots are real and unequal.

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

Answer 2

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▪Given :-

For a Quadratic Polynomial :

   

• Sum of Roots = √2

• Product of Roots = 1/4

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▪To Find :-

• The Quadratic Polynomial.

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▪Key Point :-

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

Then,

The quadratic polynomial is given by :-

$\pmb{  {x}^{2}  - s \: x + p}$

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▪Solution :-

Here,

• Sum = s = √2

• Product = p = 1/4

So,

Required Polynomial should be

$\bf{x}^{2}  - \sqrt{ 2 \:}x + \dfrac{1}{4}$

$\Large\pink{ :\longmapsto\pmb{4  {x}^{2}  - 4 \sqrt{2}\:x +1}}$

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

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