Question

Find the zroes of 3ײ -× -4 and verify the relationship between the zeroes and its coefficients.​

Answer 1

Answer:

Given :-

• 3x² - x - 4

To Find :-

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

Solution :-

Given Equation :

$\bigstar\: \: \bf 3x^2 - x - 4$

Let,

$\implies \sf p(x) =\: 3x^2 - x - 4$

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

By putting p(x) = 0 we get,

$\implies \sf 3x^2 - x - 4 =\: 0$

$\implies \sf 3x^2 - (4 - 3)x - 4 =\: 0$

$\implies \sf 3x^2 - 4x + 3x - 4 =\: 0\: \: \bigg\lgroup \sf\bold{\pink{By\: splitting\: middle\: term}}\bigg\rgroup$

$\implies \sf x(3x - 4) + 1(3x - 4) =\: 0$

$\implies \sf (3x - 4)(x + 1) =\: 0$

$\implies \bf 3x - 4 =\: 0$

$\implies \sf 3x =\: 4$

$\implies \sf\bold{\red{x =\: \dfrac{4}{3}}}$

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

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

${\small{\bold{\underline{\therefore\: The\: zeroes\: of\: quadratic\: polynomial\: are\: \dfrac{4}{3}\: and\: - 1\: respectively\: .}}}}$

Hence,

• α = 4/3
• β = - 1

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

Given Equation :

$\bigstar\: \: \bf 3x^2 - x - 4$

By comparing with ax² + bx + c we get,

• a = 3
• b = - 1
• c = - 4

❒ Sum Of Zeroes :

As we know that :

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

According to the question by using the formula we get,

$\implies \sf \dfrac{4}{3} + (- 1) =\: \dfrac{- (- 1)}{3}$

$\implies \sf \dfrac{4}{3} - 1 =\: \dfrac{1}{3}$

$\implies \sf \dfrac{4 - 3}{3} =\: \dfrac{1}{3}$

$\implies \sf\bold{\purple{\dfrac{1}{3} =\: \dfrac{1}{3}}}$

Hence, Verified.

❒ Product Of Zeroes :

As we know that :

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

According to the question by using the formula we get,

$\implies \sf \dfrac{4}{3} \times (- 1) =\: \dfrac{- 4}{3}$

$\implies \sf\bold{\purple{\dfrac{- 4}{3} =\: \dfrac{- 4}{3}}}$

Hence, Verified.