Question

If the sum of the roots of $kx^{2} + 2x + 3k = 0$ is equal to their product, then $k =$

A $\frac{3}{2}$

B $\frac{2}{3}$

C $\frac{-2}{3}$

D $\frac{-3}{2}$

✨ Free AI Tools for You

Powered by Chatterbot AI

Answer

To solve the problem of finding ( k ) for which the sum of the roots of the quadratic equation ( kx^2 + 2x + 3k = 0 ) is equal to their product, follow these steps:

  1. Identify the coefficients: Given the quadratic equation ( kx^2 + 2x + 3k = 0 ), we can compare it with the standard form ( ax^2 + bx + c = 0 ).

    • ( a = k )
    • ( b = 2 )
    • ( c = 3k )
  2. Use the properties of quadratic roots:

    • The sum of the roots ( \alpha ) and ( \beta ) of a quadratic equation ( ax^2 + bx + c = 0 ) is given by ( \alpha + \beta = -\frac{b}{a} ).
    • The product of the roots is ( \alpha \cdot \beta = \frac{c}{a} ).
  3. Calculate the sum and product:

    • The sum of the roots is:

    $$ \alpha + \beta = -\frac{b}{a} = -\frac{2}{k} $$

    • The product of the roots is:

    $$ \alpha \cdot \beta = \frac{c}{a} = \frac{3k}{k} = 3 $$

  4. Set the sum equal to the product:

    Because we are given that the sum of the roots equals their product, we set the equations equal to each other:

    $$ -\frac{2}{k} = 3 $$

  5. Solve for ( k ):

    • Multiply both sides by ( k ):

    $$ -2 = 3k $$

    • Divide both sides by 3:

    $$ k = -\frac{2}{3} $$

Thus, the value of ( k ) is [ \boxed{-\frac{2}{3}} ].


Was this helpful?

India's 1st AI Doubt Solver for CBSE, JEE, and NEET

Ask a Question for Free

and then it's just ₹212 a month