If α, β are zeros of x² − 6x + k, what is the value of k if 3α + 2β = 20?
2019
If α, β are zeros of x² − 6x + k, what is the value of k if 3α + 2β = 20?
- A.
8
- B.
-8
- C.
-2
- D.
-16
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Correct answer: D
Key facts: For the quadratic x² − 6x + k, the sum of the roots α + β = 6 and the product αβ = k.
Use the given equation 3α + 2β = 20 and substitute β = 6 − α into it.
Compute: 3α + 2(6 − α) = 20 ⇒ 3α + 12 − 2α = 20 ⇒ α + 12 = 20 ⇒ α = 8.
Then β = 6 − α = 6 − 8 = −2.
Therefore k = αβ = 8 × (−2) = −16.
Answer: k = −16.
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