Question

Find the zeroes of the quadratic polynomial and verify the relation between the zeroes and the coefficients 9x^2-3x-2

Answer 1

Step-by-step explanation:

Comparing the equation by ax^2+bx+c

We get,

a=9 b=-3 c=-2

9x^2-3x-2

9x^2-6x+3x-2(on factorising)

3x(3x-2) +1(3x-2)

3x+1=o

so x=-1/3(alfa)

3x-2=0

so, x=2/3(beta)

Sum of zeroes=-b/a

alfa + beta=-b/a

-1/3+(2/3)=3/9

1/3=1/3

Product of zeroes=c/a

alfa*beta=c/a

-1/3*(2/3)=-2/9

-2/9=-2/9

Hence, proved

Answer 2

$\underline\mathfrak{Given:-}$

• P (x) => 9x² - 3x - 2

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

• Find the zeroes are coefficients ......?

$\underline\mathfrak{Solutions:-}$

• $\: \: \: \: \: P \: {(x)} \: \: = \: \: {9x}^{2} \: - \: {3x} \: - \: {2}$

$\: \: \: \: \: \leadsto \: \: {9x}^{2} \: - \: {3x} \: - \: {2}$

$\: \: \: \: \: \leadsto \: \: {9x}^{2} \: - \: {6x} \: + \: {3x} \: - \: {2}$

$\: \: \: \: \: \leadsto \: \: {3x} \: {({3x} \: - \: {2})} \: + \: {1} \: {({3x} \: - \: {2})}$

$\: \: \: \: \: \leadsto \: \: {({3x} \: + \: {1})} \: \: \: {({3x} \: - \: {2})}$

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

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

$\underline\mathfrak{Verification:-}$

• $\: \: \: \: \: P \: {(x)} \: \: = \: \: {9x}^{2} \: - \: {3x} \: - \: {2}$

• a = 9
• b = -3
• c = -2

$\: \: \: \: \: \therefore {Sum \: \: of \: \: zeroes} \: \: = \: \: \frac{ \: - \: coefficient \: \: of \: \: x}{coefficient \: \: of \: \: {x}^{2}}$

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

$\: \: \: \: \: \leadsto \: \: \frac{-1}{3} \: + \: \frac{2}{3} \: \: = \: \: \frac{- \: {(-3)}}{9}$

$\: \: \: \: \: \leadsto \: \: \frac{{-3} \: + \: {2}}{3} \: \: = \: \: \frac{3}{9}$

$\: \: \: \: \: \leadsto \: \: \frac{1}{3} \: \: = \: \: \cancel{\frac{3}{9}}$

$\: \: \: \: \: \leadsto \: \: \frac{1}{3} \: \: = \: \: \frac{1}{3}$

$\: \: \: \: \: \therefore {Product \: \: of \: \: zeroes} \: \: = \: \: \frac{constant \: \: term}{coefficient \: \: of \: \: {x}^{2}}$

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

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

$\: \: \: \: \: \leadsto \: \: \frac{-2}{9} \: \: = \: \: \frac{-2}{9}$

Verified.