If-the-quadratic-equation-3x-2-8x-2k-0-has-two-different-negative-real-roots-Determine-the-range-of-k-For-example-0-lt-k-lt-3-or-I-will-force-you-to-determine-

Question Number 215166 by MathematicalUser2357 last updated on 30/Dec/24

If the quadratic equation 3 x^{2} + 8 x + 2 k = 0 has two different negative real roots,

Determine the range of k (For example, 0 < k < 3) or I will force you to determine

Answered by A5T last updated on 30/Dec/24

x_{1, 2} = \frac{- 8 \underline{+} \sqrt{64 - 24 k}}{6} = \frac{- 8 \underline{+} 2 \sqrt{16 - 6 k}}{6}

= \frac{- 4 \underline{+} \sqrt{16 - 6 k}}{3};

16 - 6 k \neq 0 \land 16 - 6 k > 0 \Rightarrow k < \frac{8}{3} \land - 4 + \sqrt{16 - 6 k} < 0

\Rightarrow k < \frac{8}{3} \land 16 - 6 k ⩽ 16 \Rightarrow k > 0

\Rightarrow 0 < k < \frac{8}{3}

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