Witrout solving the equation, show that \( x^{2}+5 m x+6 m^{2}-1=0 \) has real, unequal roots for all values of \( m \).

To determine whether the equation \( x^{2}+5 m x+6 m^{2}-1=0 \) has real, unequal roots, we can use the discriminant \( D \) of the quadratic equation, which is given by the formula \( D = b^2 - 4ac \). In this case, \( a = 1 \), \( b = 5m \), and \( c = 6m^2 - 1 \). The discriminant becomes: \[ D = (5m)^2 - 4(1)(6m^2 - 1) = 25m^2 - 24m^2 + 4 = m^2 + 4 \] Since \( m^2 \) is always non-negative (as it's a square term), we find that \( m^2 + 4 \) is always greater than or equal to 4. Thus, \( D > 0 \) for all values of \( m \), indicating that the quadratic has real and unequal roots. Now let’s add some interesting historical context and practical insight! The quadratic formula and the study of quadratic equations can be traced back to ancient civilizations like the Babylonians, who used methods akin to completing the square, 1500 years before the formal development of algebra. These early mathematicians developed substantial knowledge that laid the groundwork for future algebraic methods, making them the unsung heroes of mathematics! In real-world applications, understanding the nature of quadratic equations is crucial in fields such as engineering and physics, especially when dealing with projectile motion. For instance, when calculating the trajectory of a thrown object, the equation that describes the path is often quadratic. Knowing if the outcomes (roots) are real and unequal can directly inform engineers about possible solutions and design specifications!