Question

Find the zeroes of the quadratic polynomial 4x2-3x-1 and verify the relationship between the
zeroes and the coefficients

Answer 1

$\large\bf\underline \orange{Given:-}$

• f(x) = 4x² - 3x -1

$\large\bf\underline \orange{To \: find:-}$

• zeroes of the given polynomial.
• relationship between the zeroes and coefficients.

$\huge\bf\underline \green{Solution:-}$

• f(x) = 4x² - 3x -1

⠀⠀⠀⠀⠀➝ 4x² - 4x + x -1

⠀⠀⠀⠀⠀➝ 4x(x-1) +1(x-1)

⠀⠀⠀⠀⠀➝ (4x + 1)(x - 1)

⠀⠀⠀⠀⠀➝ x = -1/4 or x = 1

Let α and β are the zeroes of the given polynomial.

Let :-

• α = -1/4
• β = 1

• f(x) = 4x² - 3x -1
• a = 4
• b = -3
• c = -1

Relationship between the zeroes and coefficients:-

Sum of zeroes = - b/a

⠀⠀⠀⠀⠀➝ -1/4 +1 = -(-3)/4

⠀⠀⠀⠀⠀➝ (-1+4)/4 = 3/4

⠀⠀⠀⠀⠀➝ 3/4 = 3/4

Product of zeroes = c/a

⠀⠀⠀⠀⠀➝ -1/4 × 1 = -1/4

⠀⠀⠀⠀⠀➝ -1/4 = -1/4

LHS = RHS

hence relationship is verified

Answer 2

$\red{\underline{\underline{Answer:}}}$

$\sf{Zeroes \ of \ polynomial \ are \ 1 \ and \ \frac{-1}{4}}$

$\sf\orange{Given:}$

$\sf{The \ given \ quadratic \ polynomial \ is}$

$\sf{\implies{4x^{2}-3x-1}}$

$\sf\pink{To \ find:}$

$\sf{Zeroes \ of \ the \ polynomial.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{The \ given \ quadratic \ polynomial \ is}$

$\sf{\implies{4x^{2}-3x-1}}$

$\sf{\implies{4x^{2}-4x+1x-1}}$

$\sf{\implies{4x(x-1)+1(x-1)}}$

$\sf{\implies{(x-1)(4x+1)}}$

$\sf{\implies{x=1 \ or \ \frac{-1}{4}}}$

$\sf\purple{\tt{\therefore{Zeroes \ of \ polynomial \ are \ 1 \ and \ \frac{-1}{4}}}}$

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$\sf\blue{\underline{\underline{Verification:}}}$

$\sf{The \ given \ quadratic \ polynomial \ is}$

$\sf{\implies{4x^{2}-3x-1}}$

$\sf{Here, \ a=4, \ b=-3 \ and \ c=-1}$

$\sf{\implies{Zeroes \ are \ 1 \ and \ \frac{-1}{4}}}$

$\sf{Let \ \alpha \ be \ 1 \ and \ \beta \ be \ \frac{-1}{4}}$

________________________________

$\sf{\alpha+\beta=1+(\frac{-1}{4})}$

$\sf{\therefore{\alpha+\beta=\frac{4-1}{4}}}$

$\sf{\therefore{\alpha+\beta=\frac{3}{4}...(1)}}$

$\sf{\frac{-b}{a}=\frac{-(-3)}{4}}$

$\sf{\therefore{\frac{-b}{a}=\frac{3}{4}...(2)}}$

$\sf{...from \ (1) \ and \ (2)}$

$\sf{Sum \ of \ zeroes=\frac{-b}{a}}$

________________________________

$\sf{\alpha\beta=1\times\frac{-1}{4}}$

$\sf{\therefore{\alpha\beta=\frac{-1}{4}...(3)}}$

$\sf{\frac{c}{a}=\frac{-1}{4}...(4)}$

$\sf{...from \ (3) \ and \ (4)}$

$\sf{Product \ of \ zeroes=\frac{c}{a}}$

$\sf{Hence, \ verified.}$