If the product of the roots of the equation 3kx2 − 25kx + k + 8 = 0 is 3, then…

Concept

For any quadratic equation ax2 + bx + c = 0 with a ≠ 0, Vieta's relations tie the roots to the coefficients directly: the product of the two roots equals c/a, and their sum equals −b/a. A condition placed on the product of the roots is therefore just a condition on the ratio of the constant term to the leading coefficient, and it can be imposed without ever solving the equation or finding the roots themselves.

Applying it here

Compare 3kx2 − 25kx + (k + 8) = 0 with the standard form ax2 + bx + c = 0.

1. Read off the coefficients: a = 3k, b = −25k and c = k + 8. The expression is a quadratic only when the leading coefficient is non-zero, so k ≠ 0.

2. Write the product of the roots using Vieta: product = c/a = (k + 8)/(3k).

3. Impose the condition given in the question, namely that this product equals 3: (k + 8)/(3k) = 3.

4. Multiply both sides by 3k, which is legitimate because k ≠ 0: k + 8 = 9k.

5. Collect the k terms on one side: 8 = 9k − k, that is 8 = 8k.

6. Divide both sides by 8: k = 1.

Cross-check

Substituting k = 1 back into the original equation gives 3x2 − 25x + 9 = 0. Its product of roots is 9/3 = 3, exactly the value the question demands, and its discriminant (−25)2 − 4 × 3 × 9 = 625 − 108 = 517 is positive, so the equation genuinely has two real roots whose product is 3. Because step 5 reduces the whole condition to the linear equation 8k = 8, exactly one value of k can satisfy it, and that value is k = 1.