Find the roots exactly: Solve 2x^2 − 11x + 15 = 0.
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A3 and 5/2
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B-3 and -5/2
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C5 and 3/2
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D-5 and -3/2
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E−3 and 5/2
Answer
Correct Answer: 3 and 5/2
Explanation
Introduction / Context:Factoring is often the fastest way to solve a quadratic with integer coefficients if a suitable factor pair exists. Here the numbers factor cleanly, yielding rational roots without the quadratic formula.
Given Data / Assumptions:
- Equation: 2x^2 − 11x + 15 = 0
- Look for factorization into (2x − m)(x − n) with mn = 15 and 2n + m = 11.
Concept / Approach:We seek integers m, n such that 2x^2 − 11x + 15 = (2x − 5)(x − 3). Then set each factor to zero to get the roots. This is quicker than the quadratic formula and equally valid.
Step-by-Step Solution:
2x^2 − 11x + 15 = (2x − 5)(x − 3)Set factors to zero: 2x − 5 = 0 ⇒ x = 5/2; x − 3 = 0 ⇒ x = 3Verification / Alternative check:Expand (2x − 5)(x − 3) = 2x^2 − 6x − 5x + 15 = 2x^2 − 11x + 15, confirming correctness.
Why Other Options Are Wrong:
- Other sign variants do not satisfy the original equation when substituted.
Common Pitfalls:Misassigning signs to factor pairs of 15, or mixing the coefficient 2 into the wrong factor, leading to cross-term errors.
Final Answer:3 and 5/2