Find the value(s) of p for which the quadratic equation 2x² + 3x + p = 0 has real roots.

For real roots, Discriminant ≥ 0

or , b² - 4ac ≥ 0

The above equation is 2x² + 3x + p = 0
Comparing with ax² + bx + c = we obtain,
a = 2 , b = 3 , c = p

Putting values in b² - 4ac ≥ 0 we get,

$= 3^2 - 4 \times 2 \times p \ge 0 \\[0.5em] = 9 - 8p \ge 0 \\[0.5em] \Rightarrow 9 \ge 8p \\[0.5em] \Rightarrow 8p \le 9 \\[0.5em] \Rightarrow p \le \dfrac{9}{8} \\[0.5em]$

Hence the value of p is ≤ $\dfrac{9}{8}$.