Question

11. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the
value of k is:

Answer 1

Question :-

If the sum of square of zeros of the quadratic polynomial 6x² + x + k = 0 is 25/36 ,then find the value of k .

Answer :-

→ k = -2

To Find :-

→ Value of k

Explanation :-

we have ,

→ 6x² + x + k = 0

Let , a and b are the zeros of this quadratic polynomial .

hence ,

$\to \boxed{ \sf{sum \: of \: zeros = - \frac{coefficient \: of \: x}{ coefficient \: of \: {x}^{2}}}} \\ \\ \to \boxed{ \sf{ a + b = \frac{ - 1}{6} }} \\ \large{ \bold{\bf{ and}} \:} \\ \\ \to \boxed{\sf{product \: of \: zeros = \frac{constant}{coefficient \: of \: {x}^{2}}}} \\ \therefore \: \\ \\ \to \boxed{ \sf{ ab = \frac{k}{6} }}$

according to the question ,

$\implies \sf{{a}^{2} + {b}^{2} = \frac{25}{36} } \\ \red{ \tt{we \: can \: write \: it \: }} \\ \\ \implies \sf{ {(a + b)}^{2} - 2ab = \frac{25}{36} } \\ \therefore \\ \implies \sf{{ \bigg( \frac{ - 1}{6} \bigg) }^{2} - 2 \times \frac{k}{6} = \frac{25}{36} } \\ \\ \implies \sf{ \frac{1}{36} - \frac{k}{ 3} = \frac{25}{36} } \\ \\ \implies \sf{ \frac{1 - 12k}{36} = \frac{25}{36} } \\ \\ \implies \sf{1 - 12k = 25} \\ \\ \implies \sf{ 12k = 1 - 25} \\ \\ \implies \sf{ 12k = - 24} \\ \\ \implies \boxed{ \sf{ k = - 2}}$

Verification :-

Now we have

→ 6x² + x -2 = 0

Split the middle term

→ 6x² +4x - 3x - 2 = 0

→ 2x(3x +2) - 1(3x +2) = 0

→ (2x-1)(3x+2) = 0

$(1) \sf\: if \: \\ \to \sf{2x - 1 = 0} \\ \: \: \\ \to \boxed{ \sf{ x = \frac{1}{2} }} \\ \\ (2) \sf \: if \: \\ \to \sf{ 3x + 2 = 0} \\ \\ \to \boxed{ \sf{x = \frac{ - 2}{3} }}$

sum square of its zeros is 25/36

$\to \: \frac{1}{4} + \frac{4}{9} = \frac{9 + 16}{36} \\ \\ \to \: \frac{25}{36}$

hence verified .

Answer 2

$\tt{\huge{\orange {Hello!!! }}} S.D$

QUESTION :

11. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:

SOLUTION :

f(x) = 6x^2 + x + k

Sum Of Zeroes = -b / a = { -1 } / { 6 }...........[ 1 ]

Product Of Zeroes = c / a = { k } / { 6 } ..........[ 2 ]

Therefore :

{ Alpha } ^ 2 +{ Beta} ^ 2 = 25 / 36

=> { Alpha + Beta } ^2 - 2 [ Alpha } { Beta } = { 25 } / { 36 }

Substituting the required Values :

{ 1 } / { 36 } - { k } { 3 } = { 25 } { 36 }

=> { k } / { 3 } = - { 24 } { 36 }

=> K = -2

Hence the value of K is -2........[ A ]