Question

If one zero of the polynomial \( (a^2 + 9)x^2 + 13x + 6a \) is reciprocal of the other, find \( a \).

Hint:

Equation Formation from Word Problems: Convert conditions into equations and solve them.

Answer 1

If roots are reciprocal:
\[ \alpha \cdot \beta = 1 \] Using the product of roots formula:
\[ \frac{c}{a} = 1 \] \[ \frac{6a}{a^2 + 9} = 1 \] \[ 6a = a^2 + 9 \] \[ a^2 - 6a + 9 = 0 \] \[ (a - 3)^2 = 0 \] \[ a = 3 \] Thus, \( a = \mathbf{3} \).
Correct Answer: \( a = 3 \)