Given equation:
The given equation is 2x^2 + 3x + k = 0, where k is a real number.

Discriminant:
To find the conditions for the equation to have two distinct real roots, we need to consider the discriminant of the quadratic equation.

The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula:
D = b^2 - 4ac

Condition for two distinct real roots:
For a quadratic equation to have two distinct real roots, the discriminant must be greater than zero (D > 0).

Applying the condition:
In the given equation, we have 2x^2 + 3x + k = 0.

Comparing this with the standard form of a quadratic equation ax^2 + bx + c = 0, we have:
a = 2, b = 3, c = k

Substituting these values into the discriminant formula, we get:
D = (3)^2 - 4(2)(k) = 9 - 8k

Condition for two distinct real roots:
To find the real number k for which the equation has two distinct real roots, we need to solve the inequality D > 0.

Substituting the value of D, we have:
9 - 8k > 0

Solving the inequality:
To solve the inequality, we isolate k on one side:
-8k > -9
k < />

Therefore, the real number k for which the equation 2x^2 + 3x + k = 0 has two distinct real roots is k < />

Explanation:
The discriminant of a quadratic equation determines the nature of its roots. If the discriminant is positive, the equation has two distinct real roots. In this case, we applied the condition for two distinct real roots by setting the discriminant greater than zero and solving the resulting inequality. The final result k < 9/8="" indicates="" that="" for="" any="" value="" of="" k="" less="" than="" 9/8,="" the="" equation="" will="" have="" two="" distinct="" real="" roots.="" 9/8="" indicates="" that="" for="" any="" value="" of="" k="" less="" than="" 9/8,="" the="" equation="" will="" have="" two="" distinct="" real="" />