What is the value of \( k \) on \( t x^{2}+3 x-9=0 \)

To find the value of \( k \) in the equation \( t x^{2} + 3 x - 9 = 0 \), we assume that \( t \) represents the coefficient of \( x^2 \). It’s essential to realize that for the quadratic equation to have real solutions, the discriminant \( D = b^2 - 4ac \) must be non-negative. Here, \( a = t \), \( b = 3 \), and \( c = -9 \). The discriminant becomes \( D = 3^2 - 4(t)(-9) = 9 + 36t \). Setting this greater than or equal to zero gives \( 36t + 9 \geq 0 \), leading to \( t \geq -\frac{1}{4} \). So, \( k \) can depend on any chosen \( t \geq -\frac{1}{4} \) for which the quadratic has real solutions. The relationship between quadratic coefficients and their roots is fascinating! This relationship gets a lot of real-world applications, including in physics for projectile motion and in economics when modeling profit scenarios. Understanding how changes in coefficients affect the roots can allow you to predict behaviors like how high a ball will go or how much profit to expect based on different pricing strategies. It’s like having a cheat sheet for understanding the world around you!