Use the discriminant to tell whether the solutions of this equation are real or complex \[ x^{2}+5 x+2=0 \]

To determine the nature of the solutions for the equation \( x^{2}+5x+2=0 \), we can use the discriminant, which is calculated using the formula \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = 5 \), and \( c = 2 \). Plugging these values into the formula gives us: \[ D = 5^2 - 4 \cdot 1 \cdot 2 = 25 - 8 = 17. \] Since the discriminant \( D \) is positive (17), this means the equation has two distinct real solutions. Now get this: a bit of mathematical history! The discriminant \( D \), introduced by mathematicians like René Descartes and later developed further, is a crucial concept not just for quadratic equations, but also in higher-degree polynomials! It helps give insight into the behavior of the solutions, whether they be real or complex. Looking for some practical insight? In real-world scenarios, understanding the nature of solutions to quadratic equations is vital, especially in fields like physics and engineering. For example, determining whether the trajectory of a projectile intersects with a target can hinge on analyzing a quadratic equation and its discriminant—helping engineers and physicists make informed decisions!