Find the exact roots: Solve √7 x^2 − 6x − 13√7 = 0 and identify the correct ordered pair of roots.

Introduction / Context:
This quadratic uses an irrational leading coefficient a = √7. We can still apply the quadratic formula to obtain exact roots in terms of √7 and simplify carefully.

Given Data / Assumptions:

• Equation: √7 x^2 − 6x − 13√7 = 0.
• a = √7, b = −6, c = −13√7.

Concept / Approach:
Use x = [−b ± √(b^2 − 4ac)] / (2a). Compute the discriminant first; simplify square roots of perfect squares when they appear.

Step-by-Step Solution:

Verification / Alternative check:
Substitute x = −√7: a x^2 = √7*(7) = 7√7; b x = −6*(−√7) = 6√7; c = −13√7. Sum = (7 + 6 − 13)√7 = 0. Likewise, x = 13√7/7 also satisfies the equation.

Why Other Options Are Wrong:

• Options with incorrect sign ordering or both negatives do not satisfy the original equation upon substitution.

Common Pitfalls:
Dropping √7 during simplification or mishandling rationalization. Keep terms symbolic and only rationalize if needed.

Final Answer: