Question

If the sum of the zeros of the polynomial 3xsquare - (3k -- 2)x - (k - 6) is
equal to the product of the roots, then find the value of k.​

Answer 1

Answer:

k = 2

Step-by-step explanation:

Given :

The sum of the zeros of the polynomial 3x² - (3k – 2)x - (k - 6) is equal to the product of the roots.

To find :

the value of k

Solution :

For a quadratic equation of the form ax² + bx + c = 0 ;

⟿ sum of roots = –(x coefficient)/x² coefficient = –b/a

⟿ product of roots = constant/x² coefficient = c/a

For the given quadratic equation,

x² coefficient, a = 3

x coefficient, b = –(3k – 2)

constant, c = –(k – 6)

Sum of zeroes = Product of zeroes

–b/a = c/a

–b = c

–[–(3k – 2)] = –(k – 6)

3k – 2 = –(k – 6)

3k – 2 = –k + 6

3k + k = 6 + 2

4k = 8

k = 8/4

k = 2

The value of k is 2

Answer 2

${\boxed{\underline{\tt{\orange{Required \: \: answer:-}}}}}$

✿ k = 2

★GIVEN:-

• $\displaystyle\sf{3x^2 - (3k -- 2)x - (k - 6)}$

★TO FIND:-

• $\displaystyle \sf{Value \: \: of \: \: K }$

★FORMULA USED:-

• $\displaystyle\sf{Sum\: \: of \:\:roots \:=\: \frac{ - b}{a}}$

• $\displaystyle\sf{product\: \: of \:\:roots \:=\: \frac{ c}{a}}$

★SOLUTION:-

$: \longrightarrow\displaystyle\sf{3x^2 - (3k -- 2)x - (k - 6) = 0}$

• a = 3

• b = -(3k-2)

• c = -(k-6)

$\displaystyle\sf{Sum\: \: of \:\:roots \:=\: \frac{ - b}{a} = \frac{ - ( - (3k - 2))}{3} } \\ \\ \displaystyle\sf{product\: \: of \:\:roots \:=\: \frac{ c}{a} = \frac{ -(k - 6)}{3} } \\ \sf{For \: the \: given \: problem:- }Sum \: \: of \: \: roots = Product \: \: of \: \: roots \\ \\ : \longrightarrow \displaystyle \sf{ \frac{(3k - 2)}{3} = \frac{ - (k - 6)}{3} } \\ :\longrightarrow \displaystyle \sf{ 3k - 2 = - k + 6 } \\ : \longrightarrow \displaystyle \sf{ 3k + k = 2 + 6 } \\ : \longrightarrow \displaystyle \sf{ 4k = 8 } \\ \\ { \boxed{ \huge{ \underline{ \underline{ \sf{ \gray{ \rm{k = 2}}}}}}}}$