Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficient. x square –2​

Answer 1

Step-by-step explanation:

Given :-

x²-2

To find :-

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients ?

Solution :-

Finding zeroes :-

Given Quadratic Polynomial is x²-2

Let P(x) = x²-2

=> P(x) =x²-(√2)²

=> P(x) = (x+√2)(x-√2)

Since (a+b)(a-b) = a²-b²

To get zeroes we write P(x) = 0

=> (x+√2)(x-√2) = 0

=> x+√2= 0 or x-√2 = 0

=> x = -√2 or x =√2

Zeroes are √2 and -√2

Relationship between the zeroes and the coefficients:-

The zeroes are √2 and -√2

Let α = √2 and β = -√2

On Comparing P(x) with the standard quadratic Polynomial ax²+bx+c

We have

a = 1

b = 0

c = -2

i) Sum of the zeroes = α+β

=> α+β

=√2-√2

= 0

= 0/1

= - ( Coefficient of x )/Coefficient of x²

= -b/a

α+ β = -b/a

ii) Product of the zeroes = αβ

=> (√2)(-√2)

= -2

= -2/1

= Constant term/ Coefficient of x²

= c/a

αβ = c/a

we get

Sum of the zeroes = -b/a

Product of the zeroes = c/a

Verified the relationship between the zeroes and the coefficients of P(x).

Answer:-

The zeroes are √2 and -√2

Used formulae:-

• The standard quadratic Polynomial ax²+bx+c

• Sum of the zeroes = -b/a

• Product of the zeroes = c/a

• (a+b)(a-b) = a²-b²

Answer 2

Given Polynomial: x² – 2.

We've to find out the zeroes of the given Quadratic polynomial x² – 2. & also, verify the relationship b/w it's zeroes and coefficients.

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$:\implies\sf x^2 - 2=0\\\\\\:\implies\sf \Big\{x\Big\}^{2} - \Big\{\sqrt{2}\Big\}^2=0\\\\\\:\implies\sf\Big\{x + \sqrt{2}\Big\} \:\Big\{x - \sqrt{2}\Big\}=0\\\\\\:\implies\underline{\boxed{\pmb{\frak{\red{x = - \sqrt{2} \;or\;\sqrt{2}}}}}}\;\bigstar$

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∴ Hence, the zeroes of the given polynomial are, α = – √2 & β = √2 respectively.

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V E R I F I C A T I O N :

★ On Comparing the Quadratic Polynomial with (ax² + bx + c = 0) —

• a = 1
• b = 0
• c = –2

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¤ Let's verify, the relationship b/w the zeroes and the coefficients —

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${\qquad\maltese\:\:\textsf{Sum of Zeroes :}} \\\\\dashrightarrow\sf \alpha + \beta = \dfrac{-b}{\;a}\\\\\\\dashrightarrow\sf - \sqrt{2} + \sqrt{2}=\dfrac{0}{1}\\\\\\\dashrightarrow{\pmb{\sf{0 = 0}}}$⠀⠀⠀

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${\qquad\maltese\:\:\textsf{Product of Zeroes :}} \\\\\dashrightarrow\sf \alpha \;\beta = \dfrac{c}{a}\\\\\\\dashrightarrow\sf -\sqrt{2} \;\sqrt{2} = \dfrac{-2}{1}\\\\\\\dashrightarrow{\pmb{\sf{-2 = -2}}}$

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⠀⠀⠀⠀⠀⠀$\qquad\therefore{\underline{\textsf{\textbf{Hence, Verified!}}}}$⠀⠀