Question

Find a quadratic polynomial each with the given numbers as the sum and product of its
zeroes respectively.
(ii) √12,1/3​

Answer 1

EXPLANATION.

$\sf \: sum \: of \: zeroes \: of \: quadratic \: equation \: = \sqrt{12}$

$\sf \: products \: of \: zeroes \: of \: quadratic \: equation \: = \dfrac{1}{3}$

As we know that,

Formula of quadratic equation.

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

Sum of zeroes of quadratic equation.

$\sf \: \alpha + \beta = \dfrac{ - b}{a}$

$\sf \: \alpha + \beta = \sqrt{12}$

Products of zeroes of quadratic equation.

$\sf \: \alpha \beta = \dfrac{c}{a}$

$\sf \: \alpha \beta = \dfrac{1}{3}$

Put the value in equation, we get.

$\sf \: {x}^{2} - ( \sqrt{12} )x + \dfrac{1}{3} = 0$

$\sf \: 3 {x}^{2} - 3 \sqrt{12} x + 1 = 0$

Answer 2

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

● Sum of quadratic polynomial = √12

● Product of quadratic polynomial = 1/3

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

● Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively √12 and 1/3.

$\bigstar \; \; \; \; \;{\large{\bold{\rm{\underline{Using \; concept}}}}}$

● Sum of zeros of any quadratic equation is given by what ?

● Product of zeros of any quadratic equation is given by what ?

● Formula to find quadratic polynomial equation.

$\bigstar \; \; \; \; \;{\large{\bold{\rm{\underline{Using \; formula}}}}}$

● 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

● Formula to find quadratic polynomial equation is x²-(α+β)x + αβ

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

~ According to the question,

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

➝ α+β = √12

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

➝ αβ = 1/3

◉ Formula to find quadratic polynomial equation is x²-(α+β)x + αβ

➝ x² - (√12)x + 1/3 = 0

• (÷ = ×) ; (× = ÷) ; (+ = -) ; (- = +)

➝ 3x² - 3√12 + 1 = 0

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

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