Solve the quadratic completely: Find the roots of 2x^2 − 9x − 18 = 0.

Introduction / Context:We solve a quadratic equation using the quadratic formula or by factoring if possible. The discriminant indicates whether the roots are real and helps simplify radicals when they are perfect squares.

Given Data / Assumptions:

• Equation: 2x^2 − 9x − 18 = 0
• a = 2, b = −9, c = −18

Concept / Approach:Use the quadratic formula x = [−b ± √(b^2 − 4ac)]/(2a). Since the discriminant is a perfect square, the roots will be rational and can be presented in fractional form.

Step-by-Step Solution:

Verification / Alternative check:Substitute x = 6: 2*36 − 54 − 18 = 0. Substitute x = −3/2: 2*(9/4) + (27/2) − 18 = 0. Both satisfy the equation.

Why Other Options Are Wrong:

• 3/2 and 6; 3/2 and −6; −3/2 and −6: Each pair has at least one sign incorrect.
• −6 and 6: −6 is not a root; −3/2 is.

Common Pitfalls:Dropping the negative sign on b or mishandling the division by 2a when simplifying.

Final Answer:-3/2 and 6