Condition on Zeros of a Quadratic Polynomial

Video Explanation

Question

If the sum of the zeroes of the quadratic polynomial

\[ f(t) = kt^2 + 2t + 3k \]

is equal to their product, find the value of \(k\).

Solution

Step 1: Write the Relations Between Zeros and Coefficients

For a quadratic polynomial \(at^2 + bt + c\):

\[ \text{Sum of zeroes} = -\frac{b}{a}, \quad \text{Product of zeroes} = \frac{c}{a} \]

Here,

\[ a = k,\quad b = 2,\quad c = 3k \]

\[ \text{Sum of zeroes} = -\frac{2}{k} \]

\[ \text{Product of zeroes} = \frac{3k}{k} = 3 \]

Step 2: Use the Given Condition

Given that sum of zeroes = product of zeroes:

\[ -\frac{2}{k} = 3 \]

Step 3: Solve for \(k\)

\[ -2 = 3k \]

\[ k = -\frac{2}{3} \]

Conclusion

The required value of \(k\) is:

\[ \boxed{-\frac{2}{3}} \]

\[ \therefore \quad k = -\frac{2}{3}. \]