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Answer 1

$\underline\mathfrak\pink{Given:-}$

• $\: \: \: \: \: P \: {(x)} \: \: = \: \: {2x}^{2} \: - \: {x} \: - \: {6}$

$\underline\mathfrak\pink{To \: \: Find:-}$

• Zeroes of polynomial.
• Relationship between the zeroes are coefficients.

$\underline\mathfrak\pink{Solutions:-}$

• $\: \: \: \: \: P \: {(x)} \: \: = \: \: {2x}^{2} \: - \: {x} \: - \: {6}$

$\: \: \: \: \: \leadsto \: \: {2x}^{2} \: - \: {x} \: - \: {6}$

$\: \: \: \: \: \leadsto \: \: {2x}^{2} \: - \: {4x} \: + \: {3x} \: - \: {6}$

$\: \: \: \: \: \leadsto \: \: {2x} \: {({x} \: - \: {2})} \: + \: {3} \: {({x} \: - \: {2})}$

$\: \: \: \: \: \leadsto \: \: {({2x} \: + \: {3})} \: \: \: {({x} \: - \: {2})}$

$\: \: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: \frac{-3}{2} \: \: \: and \: \: \: {x} \: \: = \: \: {2}$

• $\: \: \: \: \: \: \: \: \: {\alpha} \: \: = \: \: \frac{-3}{2} \: \: \: and \: \: \: {\beta} \: \: = \: \: {2}$

$\underline\mathfrak\pink{Verification:-}$

$\: \: \: \: \: \therefore \: \: P \: {(x)} \: \: = \: \: {2x}^{2} \: - \: {x} \: - \: {6}$

• a = 2
• b = -1
• c = -6

$\: \: \: \: \: \therefore \: \: {\alpha} \: + \: {\beta} \: \: = \: \: \frac{-b}{a}$

$\: \: \: \: \: \leadsto \: \: \frac{-3}{2} \: + \: {2} \: \: = \: \: \frac{- \: {-1}}{2}$

$\: \: \: \: \: \leadsto \: \: \frac{{-3} \: + \: {4}}{2} \: \: = \: \: \frac{1}{2}$

$\: \: \: \: \: \leadsto \: \: \frac{1}{2} \: \: = \: \: \frac{1}{2}$

$\: \: \: \: \: \therefore \: \: {\alpha} \: {\beta} \: \: = \: \: \frac{c}{a}$

$\: \: \: \: \: \leadsto \: \: \frac{-3}{2} \: \times \: {2} \: \: = \: \: \frac{-6}{2}$

$\: \: \: \: \: \leadsto \: \: \frac{-6}{2} \: \: = \: \: \frac{-6}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: {Verified.}$

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Answer 2

Answer:

Q:- 2x² - x - 6 = 0

=> 2x² -4x +3x - 6 = 0

=> 2x(x - 2) +3(x - 2) = 0

=> (2x + 3)(x - 2) = 0

=> 2x + 3 = 0 | x - 2 = 0

=> x = -3/2 | x = 2.

=> α = -3/2 | β = 2

Two relation are:-

1)- α + β = -b / a

2)- αβ = c/a

Eq:- 2x² - x - 6

=> a = 2;

b = -1

c = -6

1)- α + β = -(-1)/2

-3/2 + 2/1 = 1/2

-3 + 4/2 = 1/2

1/2 = 1/2 verified.

2)- αβ = -6/2

-3/2 × 2 = -3

-3 = -3 verified.

HOPE IT HELPS..

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