Finding Zeroes: F(x) = X^2 + 2x + 3 Explained

Nov 8, 2025 by Admin 46 views

Hey guys! Today, we're diving into a classic math problem: finding the zeroes (or roots) of the quadratic function f(x) = x² + 2x + 3. Don't worry if that looks intimidating; we'll break it down step-by-step so it's super easy to understand. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into solving, let's quickly define what we mean by "zeroes" or "roots" of a function. The zeroes of a function are the x-values that make the function equal to zero. In other words, we're looking for the x-values that satisfy the equation f(x) = 0. For our specific function, f(x) = x² + 2x + 3, we want to find the x values when x² + 2x + 3 = 0.

Now, quadratic functions like this one (where the highest power of x is 2) can have up to two real zeroes. They can also have no real zeroes, which we'll discover shortly. The zeroes represent the points where the parabola (the graph of the quadratic function) intersects the x-axis.

Why is finding zeroes important? Well, zeroes are crucial in many real-world applications, from physics and engineering to economics and computer science. They help us determine equilibrium points, optimal values, and points of stability in various systems. For example, in physics, the zeroes of a projectile's trajectory equation can tell us when the projectile hits the ground. In economics, they might represent break-even points for a business.

So, understanding how to find the zeroes of a function is a fundamental skill in mathematics and its applications. In the context of this example, finding the zeroes will help us understand the nature of the quadratic equation and its graph. Moreover, we will determine if the given quadratic equation has real number solutions or not.

Methods for Finding Zeroes

There are several methods we can use to find the zeroes of a quadratic function. The two most common ones are:

Factoring: This method involves rewriting the quadratic expression as a product of two linear expressions. If we can factor the quadratic, we can then set each linear expression equal to zero and solve for x.
Quadratic Formula: This formula provides a direct solution for the zeroes of any quadratic function, regardless of whether it can be factored easily. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
Completing the Square: This is another algebraic technique that transforms the quadratic expression into a perfect square trinomial, making it easier to solve for x. While it's less commonly used than the quadratic formula, it's a valuable method to understand.

In some cases, we might also be able to find the zeroes graphically by plotting the function and observing where it intersects the x-axis. However, this method is generally less precise and relies on visual estimation.

Choosing the right method depends on the specific quadratic function. If the quadratic expression is easily factorable, factoring is usually the quickest and simplest approach. However, if the quadratic is not easily factorable, the quadratic formula is the most reliable method.

For our function, f(x) = x² + 2x + 3, let's first try factoring. However, it's not immediately obvious how to factor this expression. Therefore, we'll proceed with the quadratic formula, which always works!

Applying the Quadratic Formula

Alright, let's use the quadratic formula to find the zeroes of f(x) = x² + 2x + 3. Remember, the quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 1, b = 2, and c = 3. Plugging these values into the formula, we get:

x = (-2 ± √(2² - 4 * 1 * 3)) / (2 * 1)

Now, let's simplify this expression step-by-step:

x = (-2 ± √(4 - 12)) / 2

x = (-2 ± √(-8)) / 2

Here's where things get interesting! We have a negative number under the square root (√(-8)). This means the zeroes are complex numbers, not real numbers. Why? Because the square root of a negative number is not a real number. It involves the imaginary unit, 'i', where i² = -1.

So, we can rewrite √(-8) as √(8 * -1) = √(8) * √(-1) = 2√2 * i.

Substituting this back into our equation, we get:

x = (-2 ± 2√2 * i) / 2

Now, we can simplify by dividing both terms in the numerator by 2:

x = -1 ± √2 * i

Therefore, the zeroes of the function f(x) = x² + 2x + 3 are:

x₁ = -1 + √2 * i x₂ = -1 - √2 * i

These are complex conjugate pairs. This result tells us that the parabola defined by the function f(x) = x² + 2x + 3 does not intersect the x-axis at any real point. It exists entirely above the x-axis.

Interpreting the Results

So, what does it mean that the zeroes are complex numbers? It means that the graph of the function f(x) = x² + 2x + 3 does not intersect the x-axis. Think about it: the x-axis represents all the real numbers. If the function only equals zero for complex numbers, it never actually crosses the real number line (the x-axis).

In graphical terms, the parabola opens upwards (since the coefficient of x² is positive) and its vertex is above the x-axis. Therefore, it never dips down far enough to touch or cross the x-axis.

Another way to determine this without fully solving the quadratic formula is to examine the discriminant. The discriminant is the part of the quadratic formula under the square root: b² - 4ac. In our case, the discriminant is 2² - 4 * 1 * 3 = 4 - 12 = -8. Since the discriminant is negative, the quadratic equation has no real roots. If the discriminant were positive, there would be two distinct real roots. If the discriminant were zero, there would be one real root (a repeated root).

Understanding the nature of the zeroes (real or complex) helps us visualize the graph of the quadratic function and understand its behavior. It's also crucial in various mathematical and engineering applications where the nature of the solutions can have significant implications.

Completing the Square (Alternative Method)

Just for fun, let's take a quick look at how we could have approached this problem using the "completing the square" method. While we already know the zeroes are complex, this method can be useful for understanding the structure of the quadratic.

Starting with f(x) = x² + 2x + 3, we want to rewrite it in the form (x + h)² + k. To do this, we focus on the x² + 2x part. We know that (x + 1)² = x² + 2x + 1. So, we can rewrite our function as:

f(x) = (x² + 2x + 1) + 2

f(x) = (x + 1)² + 2

Now, we have the function in the form (x + 1)² + 2. To find the zeroes, we set f(x) = 0:

(x + 1)² + 2 = 0

(x + 1)² = -2

Taking the square root of both sides, we get:

x + 1 = ±√(-2)

x = -1 ± √(-2)

x = -1 ± √2 * i

As you can see, we arrive at the same complex zeroes as we did with the quadratic formula! Completing the square provides a different perspective on solving quadratic equations and can be particularly useful in certain contexts, such as finding the vertex of a parabola.

Conclusion

So, there you have it! We've successfully found the zeroes of the function f(x) = x² + 2x + 3. And, more importantly, we discovered that the zeroes are complex numbers, meaning the graph of the function doesn't intersect the x-axis. Remember, the quadratic formula is your best friend when factoring isn't straightforward. And don't be afraid of complex numbers; they're a fascinating part of the mathematical world! I hope this explanation was helpful and easy to follow. Keep practicing, and you'll become a master at finding zeroes of functions in no time!