Question

Find the value k such that the equation
(2x+3)x square + 2[K+3]2 + (K+5) = 0 has Equal roots​

Answer 1

Given :-

• A quadratic equation,

(2k + 3)x² + 2(k + 3)x + (k + 5) = 0

To Find :-

• Value of k such that the given quadratic equation has equal roots.

Solution :-

Roots of a quadratic equation are the values that satisfies that quadratic equation.

Nature of roots can be found out by discriminant.

Which is equal to, b² - 4ac of a quadratic equation ax² + bx + c = 0

Given us,

⇒ (2k + 3)x² + 2(k + 3)x + (k + 5) = 0

Now, We have to find the value of k such that the quadratic equation has equal roots which means the discriminant is equal to 0.

Here,

• a = (2k + 3)
• b = 2(k + 3)
• c = (k + 5)

So,

⇒ b² - 4 a c = 0

⇒ { 2(k + 3) }² - 4(2k + 3)(k + 5) = 0

⇒ (2k + 6)² - 4(2k² + 10k + 3k + 15) = 0

⇒ 4k² + 36 + 24k - 8k² - 52k - 60 = 0

⇒ -4k² - 28k - 24 = 0

⇒ 4k² + 28k + 24 = 0

⇒ k² + 7k + 6 = 0

⇒ k² + 6k + k + 6 = 0

⇒ k(k + 6) + 1(k + 6) = 0

⇒ (k + 6)(k + 1) = 0

∴ k = -6, -1

Answer 2

Step-by-step explanation:

ANSWER ✍️

Given :-

A quadratic equation,

(2k + 3)x² + 2(k + 3)x + (k + 5) = 0

To Find :-

Value of k such that the given quadratic equation has equal roots.

Solution :-

Roots of a quadratic equation are the values that satisfies that quadratic equation.

Nature of roots can be found out by discriminant.

Which is equal to, b² - 4ac of a quadratic equation ax² + bx + c = 0

Given us,

⇒ (2k + 3)x² + 2(k + 3)x + (k + 5) = 0

Now, We have to find the value of k such that the quadratic equation has equal roots which means the discriminant is equal to 0.

Here,

a = (2k + 3)

b = 2(k + 3)

c = (k + 5)

So,

⇒ b² - 4 a c = 0

⇒ { 2(k + 3) }² - 4(2k + 3)(k + 5) = 0

⇒ (2k + 6)² - 4(2k² + 10k + 3k + 15) = 0

⇒ 4k² + 36 + 24k - 8k² - 52k - 60 = 0

⇒ -4k² - 28k - 24 = 0

⇒ 4k² + 28k + 24 = 0

⇒ k² + 7k + 6 = 0

⇒ k² + 6k + k + 6 = 0

⇒ k(k + 6) + 1(k + 6) = 0

⇒ (k + 6)(k + 1) = 0

∴ k = -6, -1

hope this helps you!!

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