Exploring the Solutions of Quadratic Equations

Let's delve into the fascinating world of quadratic equations together! Quadratic equations are not just abstract concepts; they have real-world applications and are essential in various fields such as physics, engineering, and economics. Here are some engaging questions designed to help you explore this topic further.

1. What is the standard form of a quadratic equation?
The standard form of a quadratic equation is expressed as
\[ ax^2 + bx + c = 0 \]
where "a," "b," and "c" are constants, and "a" cannot be zero. Understanding this form helps in identifying the coefficients that will be used in solving the equation.

2. Solve the quadratic equation x² - 5x + 6 = 0 using the quadratic formula.
To solve the given equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, a = 1, b = -5, and c = 6. Plugging in these values:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} \]
This simplifies to:
\[ x = \frac{5 \pm \sqrt{25 - 24}}{2} \]
\[ x = \frac{5 \pm 1}{2} \]
Thus, the solutions are:
\[ x = 3 \quad \text{and} \quad x = 2 \]
These roots tell us where the parabola intersects the x-axis, which is crucial in many real-life contexts, such as determining the optimal price for products.

3. Discuss the significance of the discriminant in determining the nature of roots in a quadratic equation.
The discriminant, given by
\[ D = b^2 - 4ac \]
helps us determine the nature of the roots. If D > 0, there are two distinct real roots. If D = 0, there is one real root (a repeated root). If D < 0, the roots are complex and conjugate pairs. For example, if we take the equation x² + 2x + 5 = 0, the discriminant would be
\[ D = 2^2 - 4(1)(5) = 4 - 20 = -16 \]
Since D is less than zero, this equation has complex roots, which means the parabola does not intersect the x-axis.

4. Can a quadratic equation have complex roots? Provide an example.
Yes, a quadratic equation can have complex roots. A classic example is the equation x² + 1 = 0. If we apply the quadratic formula:
\[ x = \frac{-0 \pm \sqrt{0^2 - 4(1)(1)}}{2(1)} \]
This simplifies to:
\[ x = \frac{\pm \sqrt{-4}}{2} = \frac{\pm 2i}{2} \]
Thus, the roots are
\[ x = i \quad \text{and} \quad x = -i \]
These roots indicate that there are no real solutions, and they exist in the complex number system.

5. How would you graph the function y = x² - 4x + 4? What does its graph reveal about its roots?
To graph the function, we first recognize that it can be factored as
\[ y = (x - 2)^2 \]
This indicates that the vertex of the parabola is at the point (2, 0), and since it is a perfect square, the graph touches the x-axis at this point. The graph reveals that the equation has a double root at x = 2, meaning it only touches the x-axis without crossing it. This is essential in optimization problems, such as maximizing profits or minimizing costs in business.

Engaging with each of these questions not only strengthens your understanding of quadratic equations but also highlights their relevance in various fields. Keep exploring and practicing, and you will gain confidence in solving and applying quadratic equations!