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

To determine whether the solutions of the equation \( x^{2}+5 x+2=0 \) are real or complex, we can use the discriminant, which is given by \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = 5 \), and \( c = 2 \). Calculating the discriminant: \[ D = 5^2 - 4 \cdot 1 \cdot 2 = 25 - 8 = 17 \] Since the discriminant \( D = 17 \) is greater than zero, this indicates that the equation has two distinct real solutions. Understanding the role of the discriminant can really spice things up! Each part of the quadratic equation plays a vital role in the dynamics of its solutions. Specifically, the value of the discriminant tells us if we get two distinct real solutions, one repeated real solution, or two complex solutions. If you’re looking to deepen your knowledge about quadratics, a fantastic starting point is “Algebra and Trigonometry” by Michael Sullivan. This book beautifully explores the world of polynomials, providing countless examples and exercises to foster your understanding of not just quadratics, but algebra as a whole!