Roots Of Quadratic Equation X² - 3x + 4 = 0: Explained

Hey guys! Today, we're diving into the world of quadratic equations, specifically looking at how to determine the nature of the roots for the equation x² - 3x + 4 = 0. This might sound a bit intimidating, but trust me, we'll break it down step by step so it’s super easy to understand. We will explore the discriminant, a key player in this process, and see how it helps us classify the roots. So, let’s get started and unravel this mathematical mystery together!

What are Quadratic Equations?

First things first, let’s recap what quadratic equations actually are. In simple terms, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the equation would become linear, not quadratic. Now that we have the basics down, let’s see how this applies to our specific equation.

Identifying Coefficients

Looking at our equation, x² - 3x + 4 = 0, we need to identify the coefficients 'a', 'b', and 'c'. This is pretty straightforward:

• 'a' is the coefficient of x², which in this case is 1 (since x² is the same as 1x²).
• 'b' is the coefficient of x, which is -3.
• 'c' is the constant term, which is 4.

These coefficients are crucial because they’re the building blocks we’ll use to figure out the nature of the roots. So, remember these values: a = 1, b = -3, and c = 4. Now, let's move on to the discriminant, the real game-changer in determining the root nature.

The Discriminant: Your Key to Root Nature

The discriminant is the star of the show when it comes to figuring out the nature of the roots of a quadratic equation. It’s a simple calculation, but it gives us a ton of information. The discriminant, often denoted by the Greek letter Delta (Δ), is calculated using the formula:

Δ = b² - 4ac

This formula might look a bit intimidating at first, but it’s really just plugging in the values of 'a', 'b', and 'c' that we identified earlier. Once we calculate the discriminant, its value will tell us whether the roots are real, distinct, equal, or even complex. Think of the discriminant as a kind of mathematical detective, giving us clues about the roots without actually solving the equation.

Calculating the Discriminant for x² - 3x + 4 = 0

Let's put this formula to work for our equation, x² - 3x + 4 = 0. We already know that:

• a = 1
• b = -3
• c = 4

Now, we just plug these values into the discriminant formula:

Δ = (-3)² - 4 * 1 * 4

Let's break it down step by step:

1. (-3)² = 9
2. 4 * 1 * 4 = 16
3. So, Δ = 9 - 16
4. Therefore, Δ = -7

So, the discriminant for our equation is -7. This negative value is super important because it’s going to tell us a lot about the nature of the roots. Now, let's interpret what this means.

Interpreting the Discriminant: What Does Δ = -7 Mean?

The value of the discriminant, Δ, gives us direct insight into the nature of the roots of a quadratic equation. There are three main scenarios to consider:

1. Δ > 0 (Discriminant is positive): The equation has two distinct real roots. This means there are two different solutions for 'x', and they are both real numbers.
2. Δ = 0 (Discriminant is zero): The equation has two equal real roots. In other words, there is exactly one real solution for 'x' (a repeated root).
3. Δ < 0 (Discriminant is negative): The equation has two complex roots. This means the solutions for 'x' involve imaginary numbers (containing the square root of -1).

Now, let’s apply this to our situation. We calculated that the discriminant for x² - 3x + 4 = 0 is -7. So, what does this mean?

The Verdict: Complex Roots

Since our discriminant (Δ) is -7, which is less than 0, we can definitively say that the quadratic equation x² - 3x + 4 = 0 has two complex roots. These roots will involve imaginary numbers because the square root of a negative number is imaginary. This tells us that the graph of this quadratic equation will not intersect the x-axis, as real roots correspond to the points where the graph crosses the x-axis. Understanding this is key to visualizing the behavior of quadratic functions.

Complex Roots: A Quick Dive

Since we’ve established that our equation has complex roots, let’s briefly touch on what that means. Complex roots come in pairs and are of the form a + bi and a - bi, where:

• 'a' is the real part
• 'b' is the imaginary part
• 'i' is the imaginary unit, defined as the square root of -1 (√-1)

These roots are not real numbers and cannot be plotted on a standard number line. They exist in the complex number plane, which has both a real and an imaginary axis. While we won’t go into the full calculation of these roots today, it’s important to understand that they exist and what their general form looks like. Knowing this gives you a more complete picture of the solutions to quadratic equations.

Why Does This Matter? Real-World Applications

You might be wondering,