Solve the quadratic completely: Find the roots of 2x^2 − 9x − 18 = 0.
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A3/2 and 6
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B-3/2 and -6
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C-3/2 and 6
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D3/2 and -6
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E−6 and 6
Answer
Correct Answer: -3/2 and 6
Explanation
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:
D = b^2 − 4ac = (−9)^2 − 4*2*(−18) = 81 + 144 = 225√D = √225 = 15x = [9 ± 15]/(2*2) = (9 ± 15)/4Roots: x = 24/4 = 6 and x = −6/4 = −3/2Verification / 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