Question

If \(\alpha,\beta\) are the zeroes of the polynomial \(x^2-78x+k\) and if \(\alpha=\beta-6\), then \(k=\)

• A. 1496
• B. 1512
• C. 1572
• D. 1590

Hint:

When a relation between roots is given, first use the sum of roots to determine each root and then use the product formula.

Answer 1

The Correct Option is B

Concept: For the quadratic polynomial \[ x^2-78x+k, \] the sum and product of roots are \[ \alpha+\beta=78 \] and \[ \alpha\beta=k. \]

Step 1: Use the relation between roots.
Given, \[ \alpha=\beta-6 \] Substituting into \(\alpha+\beta=78\), \[ (\beta-6)+\beta=78 \] \[ 2\beta-6=78 \] \[ 2\beta=84 \] \[ \beta=42 \] Therefore, \[ \alpha=42-6=36 \]

Step 2: Find \(k\).
Since \[ k=\alpha\beta \] \[ k=36\times42 \] \[ k=1512 \] \centerline{{1512}}