1. The problem is to simplify or analyze the expression $x^2 - 4x + 24$. 2. This is a quadratic expression in the form $ax^2 + bx + c$ where $a=1$, $b=-4$, and $c=24$. 3. To understand its properties, we can calculate the discriminant $\Delta = b^2 - 4ac$ to check if it factors nicely or has real roots. 4. Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 1 \times 24 = 16 - 96 = -80$$ 5. Since $\Delta < 0$, the quadratic has no real roots and cannot be factored over the real numbers. 6. The expression is always positive because the leading coefficient $a=1$ is positive and the parabola opens upwards. 7. The vertex form can be found by completing the square: $$x^2 - 4x + 24 = (x^2 - 4x + 4) + 20 = (x - 2)^2 + 20$$ 8. This shows the minimum value of the expression is 20 at $x=2$. Final answer: The quadratic $x^2 - 4x + 24$ has no real roots and its minimum value is 20 at $x=2$. It can be written as $$(x - 2)^2 + 20$$.