Question

5. Find the zeroes of the quadratic polynomial 3x^2 - x - 4 and verify the relationship between the zeroes and the coefficients.Standard:- 10Content Quality Solution Required ❎ Don't Spamming ❎

Answer 1

AnswEr:-

Zeroes of polynomial = 4/3 & -1

$\rule{200}{1}$

Given:- Polynomial:-

☛ 3x² - x - 4 = 0

Let the zeroes be α & β

⇒ 3x² - x - 4 = 0

⇒ 3x² + 3x - 4x - 4 = 0

⇒ 3x(x + 1) - 4(x + 1) = 0

⇒ (3x - 4)(x + 1) = 0

⇒ 3x = 4 or x = - 1

↠ x = 4/3 or x = - 1

∴ α = 4/3

∴ β = -1

Therefore,

$\therefore\underline{\textsf{Zeroes of polynomial = {\textbf{4/3 \& -1 }}}}$

$\rule{150}{1}$

Comparing the polynomial with ax² + bx + c we get:-

Here,

• a = 3
• b = -1
• c = 4

Verifying the relationship between zeroes & coefficients.

Relationship 1:-

☛ Sum of zeroes = -b/a

↠ (α + β) = -b/a

↠ 4/3 + (-1) = -(-1)/3

↠ 4/3 - 1 = 1/3

↠ (4 - 3)/3 = 1/3

↠ 1/3 = 1/3 [Verified!]

Relationship 2:-

☛ Product of zeroes = c/a

↠ αβ = -4/3

↠ 4/3 × (-1) = -4/3

↠ -4/3 = -4/3 [Verified!]

Answer 2

AnswEr :

$\bf{\red{\underline{\underline{\bf{Given\::}}}}}$

We have the quadratic polynomial 3x² - x - 4.

$\bf{\red{\underline{\underline{\bf{To\:find\::}}}}}$

The zeroes of the polynomial and verify the relationship between the zeroes and the coefficients.

$\bf{\red{\underline{\underline{\bf{Explanation\::}}}}}$

p(x) = 3x² - x - 4

Zero of the polynomial is p(x) = 0

So;

$\leadsto\tt{3x^{2} -x-4=0}\\\\\leadsto\tt{3x^{2} +3x-4x-4=0}\\\\\leadsto\tt{3x(x+1)-4(x+1)=0}\\\\\leadsto\tt{(x+1)(3x-4)=0}\\\\\leadsto\tt{x+1=0\:\:\:Or\:\:\:3x-4=0}\\\\\leadsto\tt{x=-1\:\:\:Or\:\:\:3x=4}\\\\\leadsto\tt{\green{x=-1\:\:\:\:Or\:\:\:\:x=\dfrac{4}{3} }}$

We have α = -1 and β = 4/3 are the zeroes of the polynomial.

Now;

As the given quadratic polynomial as compared with ax² + bx + c = 0

• a = 3
• b = -1
• c = -4

$\blacksquare\bf{\pink{\underline{\underline{\tt{Sum\:of\:the\:Zeroes\::}}}}}$

$\mapsto\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x^{2} }{Coefficient\:of\:x} }\\\\\\\mapsto\sf{-1+\dfrac{4}{3} =\dfrac{-(-1)}{3} }\\\\\\\mapsto\sf{\dfrac{-3+4}{3} =\dfrac{-(-1)}{3} }\\\\\\\mapsto\sf{\red{\dfrac{1}{3} =\dfrac{1}{3} }}$

$\blacksquare\bf{\pink{\underline{\underline{\tt{Product\:of\:the\:Zeroes\::}}}}}$

$\mapsto\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term }{Coefficient\:of\:x} }\\\\\\\mapsto\sf{-1 \times \dfrac{4}{3} =\dfrac{-4}{3} }\\\\\\\mapsto\sf{\dfrac{-4}{3} =\dfrac{-4}{3} }\\\\\\\mapsto\sf{\red{\dfrac{-4}{3} =\dfrac{-4}{3} }}$

Thus;

Relationship between the zeroes and coefficient is verified.