Question

If the sum and product of the zeroes of the polynomial ax2 – 6x + c is equal to 12 each, find the values of a and c.​

Answer 1

Step-by-step explanation:

ax^2-6x+C=0

genral eqn of quadratic is x^2-(sum of the roots)x+product of roots=0

sum of the roots =6/a

12=6/a

a=1/2

product of roots=C/a=12

C=12a=12/2=6

pls mark my ans as brainiest

Answer 2

❍ Let's consider $\alpha$ and $\beta$ be the two zeroes of Polynomial ax² - 6x + c .

Given that ,

The sum and product of the zeroes of the polynomial ax² – 6x + c is equal to 12 each.

• Sum of zeroes = Prduct of zeroes

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

Cofficients of Given Polynomial ax² - 6x + c :

• Cofficient of x² = a
• Cofficient of x = -6
• Constant Term = c

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\dag\underline {\frak{As,\:We\:know\:that\::}}$

$\qquad \qquad \longmapsto\sf{ \alpha + \beta = \dfrac{-(Cofficient \:of\:x)}{Cofficient \:of\:x^{2}}}$

Given that ,

$\star \alpha + \beta = 12$

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Cofficients \::}}$

$\qquad \qquad \longrightarrow \sf{ 12 = \dfrac{-(-6)}{a}}$

$\qquad \qquad \longrightarrow \sf{ 12 = \dfrac{6}{a}}$

$\qquad \qquad \longrightarrow \sf{ 12 a = 6}$

$\qquad \qquad \longrightarrow \sf{ a = \dfrac{\cancel {6}}{\cancel {12}}}$

⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {  a = \dfrac{1}{2}\: }}}}\:\bf{\bigstar}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\qquad \qquad \longmapsto\sf{ \alpha \times \beta = \dfrac{ Constant\:Term}{Cofficient \:of\:x^{2}}}$

Given that ,

$\star \alpha \times \beta = 12$

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Cofficients \::}}$

$\qquad \qquad \longrightarrow \sf{ 12 = \dfrac{c}{a}}$

$\qquad \qquad \longrightarrow \sf{ 12a = c}$

As, We know that :

$\qquad \star a = \dfrac{1}{2}$

And ,

$\star \alpha \times \beta = 12$

$\qquad \qquad \longrightarrow \sf{ \cancel {12} \times \dfrac{1}{\cancel {2}} = a}$

⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {  c = 6\: }}}}\:\bf{\bigstar}$

Thus ,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {  The\:value \:of\:a \:is\:\bf{\dfrac{1}{2}}}}}$

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {  The\:value \:of\:c \:is\:\bf{6}}}}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀