$$ \left. \begin{array} { c } { 2 x ^ { 2 } - 5 k x + \frac { 25 } { 2 } = 0 } \ { \Delta

Question

$$ \left. \begin{array} { c } { 2 x ^ { 2 } - 5 k x + \frac { 25 } { 2 } = 0 } \ { \Delta < 0 } \ { b ^ { 2 } - 4 a c < 0 } \end{array} ight. $$

UpStudy AI · Accepted Answer

1. Consider the quadratic equation
   $$
   2x^2 - 5kx + \frac{25}{2} = 0.
   $$

2. The discriminant $$ \Delta $$ of a quadratic equation $$ ax^2 + bx + c = 0 $$ is given by
   $$
   \Delta = b^2 - 4ac.
   $$

3. Here, $$ a = 2 $$, $$ b = -5k $$, and $$ c = \frac{25}{2} $$. Substitute these into the discriminant:
   $$
   \Delta = (-5k)^2 - 4 \cdot 2 \cdot \frac{25}{2}.
   $$

4. Simplify the expression:
   $$
   (-5k)^2 = 25k^2,
   $$
   and
   $$
   4 \cdot 2 \cdot \frac{25}{2} = 4 \cdot 25 = 100.
   $$
   So,
   $$
   \Delta = 25k^2 - 100.
   $$

5. We are given that $$ \Delta < 0 $$. Thus,
   $$
   25k^2 - 100 < 0.
   $$

6. Divide the inequality by $$ 25 $$ (since $$ 25 > 0 $$):
   $$
   k^2 - 4 < 0.
   $$

7. Add $$ 4 $$ to both sides:
   $$
   k^2 < 4.
   $$

8. Taking the square root of both sides, we obtain:
   $$
   -2 < k < 2.
   $$

So, the condition for $$ \Delta < 0 $$ is
$$
k \in (-2, 2).
$$
The condition for $$ \Delta < 0 $$ is $$ -2 < k < 2 $$.

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