Question

Find the values of 'k' for which the quadratic equation 2x + 3x2 + k = 0 has real roots

Answer 1

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Answer 2

Question:

Find the value of k for which the quadratic equation 3x² + 2x + k = 0 has real roots.

Answer:

k € (-∞,1/3]

Note:

• An equation of degree 2 is know as quadratic equation .

• Roots of an equation is defined as the possible values of the unknown (variable) for which the equation is satisfied.

• The maximum number of roots of an equation will be equal to its degree.

• A quadratic equation has atmost two roots.

• The general form of a quadratic equation is given as , ax² + bx + c = 0 .

• A quadratic equation has atmost two roots .

• The discriminant of the quadratic equation is given as , D = b² - 4ac .

• If D = 0 , then the quadratic equation would have real and equal roots .

• If D > 0 , then the quadratic equation would have real and distinct roots .

• If D < 0 , then the quadratic equation would have imaginary roots .

Solution:

The given quadratic equation is ;

3x² + 2x + k = 0

Clearly , we have ;

a = 3

b = 2

c = k

We know that ,

The quadratic equation will have real roots if its discriminant is greater than or equal to zero .

=> D ≥ 0

=> 2² - 4•3•k ≥ 0

=> 4 - 4•3k ≥ 0

=> 4(1 - 3k) ≥ 0

=> 1 - 3k ≥ 0

=> 1 ≥ 3k

=> 3k ≤ 1

=> k ≤ 1/3

=> k € (-∞,1/3]

Hence,

The required values of k are (-∞,1/3] .