Question

Find the zeros of the quadratic polynomial 6x2 - 3 - 7x and verify the
relationship between the zeroes and the coefficients.​

Answer 1

$\star\;\bold{\underline{\sf{\pink{Given\: Polynomial\: : \: 6x^2 - 3 - 7x}}}}$

⠀⠀⠀

$:\implies\sf 6x^2 - 7x - 3 = 0 \\\\\\:\implies\sf 6x^2 - 9x + 2x - 3 = 0 \\\\\\:\implies\sf 3x(2x - 3) + 1(2x - 3) = 0 \\\\\\:\implies\sf (2x - 3) (3x + 1) = 0 \\\\\\:\implies{\underline{\boxed{\frak{\pink{x = \dfrac{3}{2} \; \&\; \dfrac{-1}{\;3}}}}}}\;\bigstar$

$\sf{\therefore\; Zeroes\; of \; the \: Given \; polynomial \; are \; \dfrac{3}{2} \:\&\; \dfrac{-1 \: }{\; \: 3 \: }.}$

$\rule{250px}{.3ex}$

$\bf{\dag} \: \: \underline{\textsf{Relation b/w Coefficients \& Zeroes \: :}}$⠀⠀

⠀⠀⠀⠀⠀

${\qquad\maltese\:\:\textsf{Sum of Zeroes :}} \\\\\dashrightarrow\sf\:\:\alpha +\beta= \dfrac{ - \:b \: \: \: }{ \: \: \: a \: \: \:}\\\\\\\dashrightarrow\sf \bigg(\dfrac{3}{2}\bigg) + \bigg(\dfrac{-1}{ \: \: 3}\bigg) = - \dfrac{-7}{ \: \: 6} \\\\\\\dashrightarrow{\underline{\boxed{\frak{\dfrac{7}{6} = \dfrac{7}{6}}}}}$

⠀⠀⠀

${\qquad\maltese\:\:\textsf{Product of Zeroes :}}\\\\\dashrightarrow\sf\:\:\alpha\beta=\dfrac{c}{a}\\\\\\\dashrightarrow\sf \bigg(\dfrac{3}{2}\bigg) \times \bigg(\dfrac{-1 \: \: }{ \: \: 3 \: \: }\bigg) = \dfrac{-3 \: \: }{ \: \: 6 \: \: } \\\\\\\dashrightarrow{\underline{\boxed{\frak{\dfrac{-1}{\;2} = \dfrac{-1}{\;2}}}}}$

⠀⠀⠀

$\qquad\quad\therefore{\underline{\textsf{\textbf{Hence, Verified!}}}}$

Answer 2

Given : A polynomial 6x² - 7x - 3.

Need to find : It's zeroes and relationship between the zeroes and the coefficients.

⠀⠀⠀⠀⠀⠀____________________

Firstly, we'll find the zeroes of the quadratic polynomial.

$\: \:$

$\sf:\implies \: 6x {}^{2} - 7x - 3 = 0 \\ \\ \\ \sf:\implies \: 6x {}^{2} - 9x + 2x - 3 = 0 \\ \\ \\ \sf:\implies \: 3x(2x - 3) + 1(2x - 3) = 0 \\ \\ \\ \sf:\implies \: (3x + 1)(2x - 3) = 0 \\ \\ \\ \sf:\implies \: \underline{\boxed{\sf\purple{x = \frac{ - 1}{3}, \frac{3}{2} }}} \: \bigstar$

$\: \:$

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

$\: \:$

$\bold{ \underline{sum \: of \: zeroes \: : }}$

$\: \:$

$\sf :\implies \: α+β = \dfrac{ - b}{a} \\ \\ \\ \sf :\implies \: \frac{ - 1}{3} + \frac{3}{2} \\ \\ \\ \sf :\implies \: \frac{ - 2 + 9}{6} \\ \\ \\ \sf :\implies \: \underline{\boxed{\sf\purple{ \frac{7}{6} = \frac{7}{6} }}} \: \bigstar$

$\: \:$

$\bold{ \underline{ product \: of \: zeroes \: : }}$

$\: \:$

$\sf : \implies \: α×β = \dfrac{c}{a} \\ \\ \\ \sf : \implies \: \frac{ - 1}{3} \times \frac{3}{2} \\ \\ \\ \sf : \implies \: \underline{\boxed{\sf\purple{ \frac{ - 3}{6} = \frac{ - 3}{6} }}} \: \bigstar$

$\: \:$

$\sf\therefore{\underline{Hence, \: verified \: !}}$