Question

2. Find a quadratic polynomial each with the given numbers as the sum and product of its
zeroes respectively.
1/4 , -1
​

Answer 1

Answer

Quadratic Polynomial will be 4x²+x-4 .

Step-by-step explanation:

$\: \: \: \: \: \: \: \bull \: Sum \: of \: root \: \sf \: = \frac{1}{4}$

$\: \: \: \: \: \bull \: Product \: of \: root \: \sf \: = - 1$

We have to calculate the quadratic polynomial whose sum and product are given .

We know that a quadratic polynomial when sum and product of its zeros are are given

$\bull\bf\: f(x) \: = {x}^{2} - (sum \:of \: root)x \: + (product \: of \: root)$

now putting the value we get the required quadratic polynomial .

$\implies \sf \:f(x) \: = {x}^{2} - (\frac{1}{4})x \: + ( - 1) \: \: \\ \\ \implies \sf \: f(x) \: = {x}^{2} + \frac{1}{4}x \: - 1 \\ \\ \implies \sf \: f(x) \: = {4x}^{2} \: + \:x \: - 4 \\ \\ \\ \therefore \bf \: The \: required \: quadratic \: polynomial \: will \: be \: {4x}^{2} + x - 4$

Answer 2

Answer:

Question :-

⦿ Find a quadratic polynomial each with the given numbers as the sum and product of its

zeroes respectively : 1/4 , -1

Given :-

⦿ 1/4 , -1

Find Out :-

⦿ Find a quadratic polynomial.

Solution :-

☣ Quadratic polynomial :-

As we know that :

$\red{ \boxed{\sf{{x}^{2} - (Sum \:Of \: Root)x \: + (Product \: Of \: Root)}}}$

We have :

☢ Sum of Zeroes = ¼

☢ Product of Zeroes = - 1

According to the question or ATQ :-

$\sf \longrightarrow {x}^{2} - \bigg\{\dfrac{1}{4}\bigg\} x \: + \{ - 1\}$

$\sf \longrightarrow {x}^{2} + \dfrac{1}{4}x - 1$

$\sf \longrightarrow \boxed{\bold{\red{ 4x \: + \:x \: - 4}}}$

Henceforth, the quadratic polynomial is 4x + x - 4.

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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 Roots :

✪ b² - 4ac is the discriminant of the equation.

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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