Question

write the quadretic polynomial whose sum and product of zeroes is 3 and -2

Answer 1

EXPLANATION.

Quadratic polynomial,

Sum of zeroes of quadratic equation = 3.

Products of zeroes of quadratic equation = -2.

As we know that,

General quadratic equation = ax² + bx + c.

Sum of zeroes of quadratic equation.

⇒ α + β = -b/a.

⇒ α + β = 3.

Products of zeroes of quadratic equation.

⇒ αβ = c/a.

⇒ αβ = -2.

As we know that,

Equation of quadratic polynomial.

⇒ x² - (α + β)x + αβ.

Put the value in equation, we get.

⇒ x² - (3)x + (-2) = 0.

⇒ x² - 3x - 2 = 0.

                                                                                                                     

MORE INFORMATION.

Nature of the factors of the quadratic expression.

(1) = Real and different, if b² - 4ac > 0.

(2) = Rational and different, if b² - 4ac is a perfect square.

(3) = Real and equal, if b² - 4ac = 0.

(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.

Answer 2

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${\large{\bold{\rm{\underline{Given \; that}}}}}$

${\small{\sf{\red \bigstar Sum \: of \: zeros \: of \: quadratic \: polynomial \: is \: \bf \: 3}}}$

${\small{\sf{\red \bigstar Product \: of \: zeros \: of \: quadratic \: polynomial \: is \: \bf \: -2}}}$

${\large{\bold{\rm{\underline{To \; find}}}}}$

${\small{\sf{\red \bigstar Quadratic \: polynomial}}}$

${\large{\bold{\rm{\underline{Solution}}}}}$

${\small{\sf{\red \bigstar Quadratic \: polynomial \: is \: \bf \: x^{2} - 3x - 2 = 0}}}$

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${\large{\bold{\rm{\underline{Using \; concepts}}}}}$

${\small{\sf{\green \bigstar Sum \: of \: zeros \: of \: quadratic \: polynomial \: is \: given \: by \: what}}}$

${\small{\sf{\green \bigstar Product \: of \: zeros \: of \: quadratic \: polynomial \: is \: given \: by \: what}}}$

${\small{\sf{\green \bigstar Quadratic \: polynomial \: equation}}}$

${\large{\bold{\rm{\underline{Using \; dimensions}}}}}$

${\small{\sf{\green \bigstar Sum \: of \: zeros \: of \: quadratic \: polynomial \: is \: given \: by \: \alpha + \beta = \: -b/a}}}$

${\small{\sf{\green \bigstar Product \: of \: zeros \: of \: quadratic \: polynomial \: is \: given \: by \: \alpha \beta = \: c/a}}}$

${\small{\sf{\green \bigstar Quadratic \: polynomial \: equation \: is \: x^{2}-(\alpha + \beta)x + \alpha \beta}}}$

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${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ As we know that sum of quadratic polynomial is given by ${\small{\orange{\alpha + \beta = \: -b/a}}}$

${\small{\sf{:\implies \alpha + \beta = \: -b/a}}}$

${\small{\sf{:\implies \alpha + \beta = \: 3}}}$

~ Now as we know that sum of quadratic polynomial is given by ${\small{\orange{\alpha \beta = \: c/a}}}$

${\small{\sf{:\implies \alpha \beta = \: c/a}}}$

${\small{\sf{:\implies \alpha \beta = \: -2}}}$

~ Now as we know that the quadratic polynomial equation is ${\small{\orange{x^{2}-(\alpha + \beta)x + \alpha \beta}}}$

${\small{\sf{:\implies x^{2}-(\alpha + \beta)x + \alpha \beta}}}$

${\small{\sf{:\implies x^{2} - 3x - 2 = 0}}}$

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${\large{\bold{\rm{\underline{Additional \; knowledge}}}}}$

Knowledge about Quadratic equations -

★ Sum of zeros of any quadratic equation is given by ➝ α+β = -b/a

★ Product of zeros of any quadratic equation is given by ➝ αβ = c/a

★ A quadratic equation have 2 roots

★ ax² + bx + c = 0 is the general form of quadratic equation

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