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Answer 1

4x² - x - 5 = 0

Factorising,

(4x - 5)(x + 1) = 0

Therefore, the roots are x = 5/4 and x = -1

Sum of roots = 5/4 - 1 = 1/4

-b/a = -(-1)/4 = 1/4

Product of roots = -5/4

c/a = -5/4

Hence verified.

Answer 2

Answer:

Given :-

• 4x² - x - 5

To Find :-

• What are the zeros of quadratic polynomial and verify the relationship between the zeroes and co-efficient.

Solution :-

Given Equation :

$\bigstar\: \: \bf{4x^2 - x - 5}$

Let,

$\implies \sf p(x) =\: 4x^2 - x - 5$

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

By putting p(x) = 0 we get,

$\implies \sf 4x^2 - x - 5 =\: 0$

$\implies \sf 4x^2 - (5 - 4)x - 5 =\: 0$

$\implies \sf 4x^2 - 5x + 4x - 5 =\: 0\: \: \bigg\lgroup \sf\bold{\pink{By\: splitting\: the\: middle\: term}}\bigg\rgroup$

$\implies \sf x(4x - 5) + 1(4x - 5) =\: 0$

$\implies \sf (4x - 5)(x + 1) =\: 0$

$\longrightarrow \bf 4x - 5 =\: 0$

$\longrightarrow \sf 4x =\: 5$

$\longrightarrow \sf\bold{\red{x =\: \dfrac{5}{4}}}$

$\longrightarrow \bf x + 1 =\: 0$

$\longrightarrow \sf\bold{\red{x =\: - 1}}$

${\small{\bold{\underline{\therefore\: The\: zeroes\: of\: quadratic\: polynomial\: are\: \dfrac{5}{4}\: and\: -\: 1\: respectively\: .}}}}$

Hence we get :

• α = $\sf\dfrac{5}{4}$
• β = $\sf - 1$

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

Given Equation :

$\bigstar\: \: \bf{4x^2 - x - 5}$

By comparing with ax² + bx + c we get,

• a = 4
• b = - 1
• c = - 5

❒ 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 :

• a = 4
• b = - 1
• α = 5/4
• β = - 1

According to the question by using the formula we get,

$\longrightarrow \sf \dfrac{5}{4} + (- 1) =\: \dfrac{- (- 1)}{4}$

$\longrightarrow \sf \dfrac{5}{4} - 1 =\: \dfrac{1}{4}$

$\longrightarrow \sf \dfrac{5 - 4}{4} =\: \dfrac{1}{4}$

$\longrightarrow \sf\bold{\purple{\dfrac{1}{4} =\: \dfrac{1}{4}}}$

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 :

• a = 4
• c = - 5
• α = 5/4
• β = - 1

According to the question by using the formula we get,

$\longrightarrow \sf \dfrac{5}{4} \times (- 1) =\: \dfrac{- 5}{4}$

$\longrightarrow \sf\bold{\purple{\dfrac{- 5}{4} =\: \dfrac{- 5}{4}}}$

Hence, Verified.