Question

find zeroes of quadratic polynomial and find the relation hip between the zeroes and the coefficients 1.x2-2x-8​

Answer 1

Answer:

Given :-

• x² - 2x - 8

To Find :-

• What is the zeroes of quadratic equation.
• What is the relationship between the zeroes and co-efficient.

Solution :-

Given Equation :

$\bigstar\: \: \bf{x^2 - 2x - 8}$

Let,

$\implies \sf p(x) =\: x^2 - 2x - 8$

Zero of the polynomial is the value of x where p(x) = 0

By putting p(x) = 0, we get :

$\implies \sf x^2 - 2x - 8 =\: 0$

$\implies \sf x^2 - (4 - 2)x - 8 =\: 0$

$\implies \sf x^2 - 4x + 2x - 8 =\: 0\: \: \bigg\lgroup \sf\bold{By\: splitting\: the\: middle\: term}\bigg\rgroup$

$\implies \sf x(x - 4) + 2(x - 4) =\: 0$

$\implies \sf (x + 2)(x - 4) =\: 0$

$\longrightarrow \bf{(x + 2) =\: 0}$

$\longrightarrow \sf \bold{\red{x =\: - 2}}$

$\longrightarrow \bf{(x - 4) =\: 0}$

$\longrightarrow \sf\bold{\red{x =\: 4}}$

${\small{\bold{\underline{\therefore\: The\: zeroes\: of\: quadratic\: polynomial\: is\: -\: 2\: and\: 4\: respectively\: .}}}}$

Hence, we get :

• α = - 2
• β = 4

Now, we have to verify the relationship between the zeroes and co-efficient :

Given Equation :

$\bigstar\: \: \bf{x^2 - 2x - 8}$

By comparing with ax² + bx + c we get,

• a = 1
• b = - 2
• c = - 8

❒ Sum of Zeroes :

As we know that :

$\clubsuit$ Sum of Zeroes Formula :

$\mapsto \sf\boxed{\bold{\pink{Sum\: of\: zeroes\: (\alpha + \beta) =\: \dfrac{- b}{a}}}}$

We have :

• α = - 2
• β = 4
• a = 1
• b = - 2

According to the question by using the formula we get,

$\longrightarrow \sf - 2 + 4 =\: \dfrac{- (- 2)}{1}$

$\longrightarrow \sf 2 =\: \dfrac{2}{1}$

$\longrightarrow \sf\bold{\purple{2 =\: 2}}$

Hence, Verified.

❒ Product of Zeroes :

As we know that :

$\clubsuit$ Product of Zeroes Formula :

$\mapsto \sf\boxed{\bold{\pink{Product\: of\: zeroes\: (\alpha\beta) =\: \dfrac{c}{a}}}}$

We have :

• α = - 2
• β = 4
• a = 1
• c = - 8

According to the question by using the formula we get,

$\longrightarrow \sf - 2 \times 4 =\: \dfrac{- 8}{1}$

$\longrightarrow \sf \bold{\purple{- 8 =\: - 8}}$

Hence, Verified.

Answer 2

(i) x

2

−2x−8

Factorize the equation, we get (x+2)(x−4)

So, the value of x

2

−2x−8 is zero when x+2=0,x−4=0, i.e., when x=−2 or x=4.

Therefore, the zeros of x

2

−2x−8 are -2 and 4.

Now,

⇒Sum of zeroes =−2+4=2=−

1

2

=−

Coefficient of x

2

Coefficient of x

⇒Product of zeros =(−2)×(4)=−8 =

1

−8

=

Coefficient of

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