For the function \( f(x)=x^{2}+5 x-6 \) solve the following. \( f(x) \geq 0 \)

To solve \( f(x) \geq 0 \) for the function \( f(x) = x^{2} + 5x - 6 \), we first find the roots by setting \( f(x) = 0 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 5 \), and \( c = -6 \), we calculate: \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} = \frac{-5 \pm \sqrt{25 + 24}}{2} = \frac{-5 \pm 7}{2} \] This gives us the roots \( x = 1 \) and \( x = -6 \). The quadratic opens upwards (since the coefficient of \( x^2 \) is positive), so \( f(x) \geq 0 \) outside the roots. Thus, the solution is \( x \leq -6 \) or \( x \geq 1 \). To express this more succinctly: The solution to the inequality \( f(x) \geq 0 \) is \( x \in (-\infty, -6] \cup [1, \infty) \). Funky fact: Quadratic functions like \( f(x) \) can be visualized as parabolas! When you graph it, you’ll see how the vertex and roots influence the direction and width of these curves. Quadratics often pop up in physics for projectile motion, where the height of a thrown object over time can be described by a similar equation. If you want to dive deeper into the world of quadratics, consider checking out some algebra textbooks that focus on polynomial functions. They often contain plenty of exercises to hone your skills on solving inequalities, factoring, and determining the properties of graphed functions. Plus, there are lots of online resources and interactive tools that can help deepen your understanding!