Question

Find the zeroes of the polynomial 3x 2 – 2x – 8 and verify the relationship between the zeroes and its coefficients

Answer 1

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Answer 2

GIVEN :–

• A quadratic equation 3x² - 2x - 8 = 0.

TO FIND :–

• zeroes of the polynomial = ?

• Verify the relationship between the zeroes and its coefficients.

SOLUTION :–

$\\ \sf \implies \: \: 3 {x}^{2} - 2x - 8 = 0$

• Splitting Middle term –

$\\ \sf \implies \: \: 3 {x}^{2} - 6x + 4x - 8 = 0$

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

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

$\\ \implies \large { \boxed{ \sf x = - \dfrac{4}{3} \:, \: 2}}$

VERIFICATION :–

$\\ \sf \longrightarrow \: sum \: \: of \: \: roots \: = \dfrac{ -(coffieciant \: \: of \: \: x)}{coffieciant \: \: of \: \: {x}^{2} }$

$\\ \sf \implies \: - \dfrac{4}{3} + 2 = \dfrac{ -( - 2)}{3}$

$\\ \sf \implies \: \dfrac{6 - 4}{3} = \dfrac{2}{3}$

$\\ \sf \implies \: \dfrac{2}{3} = \dfrac{2}{3} \: \: (verified)$

$\\ \sf \longrightarrow \: product \: \: of \: \: roots \: = \dfrac{constant \: \: term}{coffieciant \: \: of \: \: {x}^{2} }$

$\\ \sf \implies \: - \dfrac{4}{3} ( 2 ) = \dfrac{ - 8}{3}$

$\\ \sf \implies \: - \dfrac{8}{3} = - \dfrac{8}{3} \: \: (verified)$