Question

find the value of k such that the equation (2k+3)x square +2(k+3)x +(k+5)=0 has equal roots

plz give answer i need it​

Answer 1

$\LARGE{ \underline{\underline{ \sf{Required \: answer:}}}}$

GiveN Quad. eq.

• (2k+3)x² + 2(k+3)x + (k+5)=0
• The equation has equal roots.

To FinD:

• Value of k here?

Step-wise-Step Explanation:

When any Quadratic equation has equal roots, the Discriminate is always equals to 0. Now Discriminate means $\rm{ {b}^{2} - 4ac}$ when the equation is compared to $\rm{a {x}^{2} + bx + c}$.

Here,

• a = 2k + 3
• b = 2k + 6
• c = k + 5

We know that D = 0. Putting the values of a, b and c from this equation.

⇒ b² - 4ac = 0

⇒ (2k + 6)² - 4(2k + 3)(k + 5) = 0

⇒ 4k² + 24k + 36 - 4(2k² + 13k + 15) = 0

⇒ 4k² + 24k + 36 - 8k² - 52k - 60 = 0

⇒ -4k² - 28k - 24 = 0

⇒ 4k² + 28k + 24 = 0

Now factorising the above quad. eq. using middle term factorisation,

⇒ 4k² + 4k + 24k + 24 = 0

⇒ 4k(k + 1) + 24(k + 1) = 0

⇒ (4k + 24)(k + 1) = 0

⇒ (k + 6)(k + 1) = 0

Equating to 0, we will get k = -6 or -1 (Ans)

Answer 2

$\huge\mathcal{\underline{\color{olive}QUESTION}}$

★ Find the value of k such that the equation (2k + 3) x² + 2(k + 3) + (k + 5) = 0 has equal roots .

$\huge{\orange{\boxed{\fcolorbox{lime}{cadetblue}{\purple{ANSWER}}}}}$

$\red\bullet\:\bf{(2k~+~3)~x^2~+~2(k~+~3)~+~(k~+~5)~=~0~}$

Here,

• a = (2k + 3)

• b = 2(k + 3)

• c = (k + 5)

᪥ As we know that,

$\longrightarrow\:\sf{D~=~b^2~-~4ac~}$

$\longrightarrow\:\sf{D~=~\Big\{2(k~+~3~)\Big\}^2~-~4\times(2k~+~3)\times(k~+~5)~}$

$\longrightarrow\:\sf{D~=~4(k^2~+~9~+~6k)~-~4\times(2k^2~+~10k~+~3k~+~15)~}$

$\longrightarrow\:\sf{D~=~4k^2~+~36~+~24k~-~8k^2~-~40k~-~12k~-~60~}$

$\longrightarrow\:\sf{D~=~4k^2~+~36~+~24k~-~8k^2~-~52k~-~60~}$

$\longrightarrow\:\sf{D~=~-~4k^2~-~24~-~28k~}$

$\longrightarrow\:\sf{D~=~-4~(k^2~+~7k~+~6)~}$

☢︎︎ The given equation will have equal roots,

• If D = 0 .

╰➝ -4 (k² + 7k + 6) = 0

╰➝ k² + 7k + 6 = 0

╰➝ k² + 6k + k + 6 = 0

╰➝ k (k + 6) + 1 (k + 6) = 0

╰➝ (k + 6) (k + 1) = 0

╰➝ k + 6 = 0 (or) k + 1 = 0

╰➝ k = -6 (or) k = -1

∴ The value of k is “ -6 or -1 ” .