Question

Find a quadratic polynomial whose sum and product respectively of those zeroes of this polynomial by factorisation
$a) \frac{ - 8}{3} , \frac{4}{3}$
$b)\frac{21}{8} , \frac{5}{16}$
$c) - 2 \sqrt{3} \: ,9$
No copied answer
Quality answer required . ​

Answer 1

EXPLANATION.

Quadratic polynomial whose sum and products of the zeroes,

As we know that,

Quadratic equation ⇒ x² - (α + β)x + αβ.

Sum of zeroes of quadratic equation.

⇒ α + β = -8/3.

Products of zeroes of quadratic equation.

⇒ αβ = 4/3.

Put the values in the equation, we get.

⇒ x² - (α + β)x + αβ = 0.

⇒ x² - (-8/3)x + 4/3 = 0.

⇒ x² + 8x/3 + 4/3 = 0.

⇒ 3x² + 8x + 4 = 0.

Sum of zeroes of quadratic equation,

⇒ α + β = 21/8.

Products of zeroes of quadratic equation,

⇒ αβ = 5/16.

Put the values in the equation, we get.

⇒ x² - (α + β)x + αβ = 0.

⇒ x² - (21/8)x + 5/16 = 0.

⇒ 16x² - 42x + 5 = 0.

Sum of zeroes of quadratic equation,

⇒ α + β = -2√3.

Products of zeroes of quadratic equation,

⇒ αβ = 9.

Put the values in equation, we get.

⇒ x² - (α + β)x + αβ = 0.

⇒ x² - (-2√3)x + 9 = 0.

⇒ x² + 2√3x + 9 = 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) = Roots are imaginary and unequal or complex conjugate, if b² - 4ac < 0.

Answer 2

General Formula :-

$\underline{\boxed{\bf Quadratic\: equation = x²-(α+β)x+αβ }}$

here, α and β are the zeroes of the question.

1st Question:-

• α+β = $\sf\dfrac{-8}{3}$

• αβ = $\sf\dfrac{4}{3}$

Quadratic equation = x² - ($\sf\dfrac{-8}{3}$)x + $\sf\dfrac{4}{3}$

$\longrightarrow$ x² + $\sf\dfrac{8}{3}$ + $\sf\dfrac{4}{3}$

$\longrightarrow$ 3x²+8x+4 (here, we have multiplied the entire equation with 3 in order to remove the denominator)

________________________

2nd question:-

• α+β = $\sf\dfrac{21}{8}$

• αβ = $\sf\dfrac{5}{16}$

Quadratic equation = x²-$\sf\dfrac{21}{8}$x + $\sf\dfrac{5}{16}$

$\longrightarrow$ 16x²-42x+5 (here, we have taken the L.C.M of the denominators and then multiplied it with the equation)

________________________

3rd Question:-

• α+β = -2√3

• αβ = 9

Quadratic equation = x²-(-2√3)x + 9

$\longrightarrow$ x²+2√3+9

________________________

Additional information:-

• Sum of zeros of a quadratic equation:- ax²+bx+c = $\sf\dfrac{-b}{a}$
• Product of zeros of a quadratic equation:- ax²+bx+c = $\sf\dfrac{c}{a}$

Nature of roots :-

Formula of Discriminant (D) is as follows:-

$\longrightarrow$ $\underline{\boxed{\bf D = b²-4ac }}$

• Real and distinct roots, when D > 0
• Real and equal roots, when D = 0
• Non - real roots, when D < 0

_________________________

hope uh r getting it :)