Question

sum and product of zeros of a quadratic polynomial are 3 and 6 respectively then find that polynomial ​

Answer 1

Answer:

Good morning

The required quadratic polynomial :

$x ^{2} - 3x + 6$

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Answer 2

Formula Used :-

Quadratic Equation Formula

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

Solution :-

◘ Given :

$\bigstar\: \: \: \sf\bold{\red{Sum\: of\: roots\: (\alpha + \beta) =\: 3}}$

Hence, the required polynomial equation are :

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

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

$\small\longrightarrow \sf x^2 - (3)x + (6) =\: 0 \\ \small\longrightarrow \sf x^2 - 3x + 6 =\: 0 \\ \small\longrightarrow \sf\bold{\red{x^2 - 3x + 6 =\: 0}}$

${\small{\bold{\underline{\therefore\: The\: polynomial\: equation\: is\: x^2 - 3x + 6 =\: 0\: .}}}}$

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❒ 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 the roots :

✪ Discriminant (D) = b² - 4ac. 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.

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