Question

find the zeros of the quadratic polynomial and verify the relationship between the zeroes and co-efficents p(x)
$4x {}^{2} - 24x + 36$
​

Answer 1

here is your answer...

$4x ^{2} - 24x + 36 = 0 \\ x ^{2} - 6x + 9 = 0 \\ x ^{2} - (3 + 3)x + 9 = 0 \\ x ^{2} - 3x - 3x + 9 = 0 \\ x(x - 3) - 3(x - 3) = 0 \\ (x - 3)(x - 3) = 0 \\ \\ zeroes \: are \: equal \: both \: \: 3$

Now Relation....

a=4

b= -24

c=36

$\alpha + \beta = \frac{ - b}{a} \\ 3 + 3 = \frac{ - ( - 24)}{4} \\ 6 = 6$

$\alpha \beta = \frac{c}{a} \\ 3 \times 3 = \frac{36}{4} \\ 9 = 9$

Thanks..

hope it helps you

Answer 2

$\underline\mathfrak{Given:-}$

• 4x² - 24x + 36

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

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

$\underline\mathfrak{Solutions:-}$

• $\: \: \: \: \: P \: {(x)} \: \: = \: \: {4x}^{2} \: - \: {24x} \: + \: {36}$

$\: \: \: \: \: \leadsto \: \: {4x}^{2} \: - \: {24x} \: + \: {36}$

$\: \: \: \: \: \leadsto \: \: {4x}^{2} \: - \: {16x} \: - \: {9x} \: + \: {36}$

$\: \: \: \: \: \leadsto \: \: {4x} \: {({x} \: - \: {4})} \: - \: {9} \: {({x} \: - \: {4})}$

$\: \: \: \: \: \leadsto \: \: {({4x} \: - \: {9})} \: \: \: {({x} \: - \: {4})}$

$\: \: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: \frac{9}{4} \: \: \: and \: \: \: {x} \: \: = \: \: {4}$

$\: \: \: \: \: \: \: \: \: {\alpha} \: \: = \: \: \frac{9}{4} \: \: \: and \: \: \: {\beta} \: \: = \: \: {4}$

$\underline\mathfrak{Verification:-}$

4x² - 24x + 36

• a = 4
• b = -24
• c = 36

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

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

$\: \: \: \: \: \leadsto \: \: \frac{9}{4} \: + \: {4} \: \: = \: \: - \: \frac{(-24)}{4}$

$\: \: \: \: \: \leadsto \: \: \frac{{9} \: + \: {16}}{4} \: \: = \: \: \frac{24}{4}$

$\: \: \: \: \: \leadsto \: \: \frac{24}{4} \: \: = \: \: \frac{24}{4}$

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

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

$\: \: \: \: \: \leadsto \: \: \frac{9}{\cancel{4}} \: \times \: \cancel{4} \: \: = \: \: \cancel{\frac{36}{4}}$

$\: \: \: \: \: \leadsto \: \: {9} \: \: = \: \: {9}$

Verified.