Graphing Quadratic Equations Solving For The Correct Solution

Hey guys! Let's dive into this math problem together and figure out which graph represents the solutions to the equation x² + 8x = -20. Don't worry, we'll break it down step by step so it's super easy to understand. So let's buckle up and figure out the correct answer.

Understanding Quadratic Equations

To decode quadratic equations, it's essential to first grasp the fundamentals of what these equations represent and how their solutions manifest graphically. Quadratic equations, characterized by the presence of a squared term (in this case, x²), plot as parabolas on a graph—those familiar U-shaped curves. Understanding the relationship between the equation's solutions and the graph is the first step to cracking this problem.

The solutions to a quadratic equation, also known as roots or zeros, are the x-values where the parabola intersects the x-axis. These are the points where the value of the equation equals zero. This is a crucial concept because it provides a visual link between the algebraic solutions and the graphical representation.

But here's a twist! Not all parabolas intersect the x-axis. Some hover above or below it, meaning they have no real solutions. Others might just touch the x-axis at a single point, indicating one real solution (a repeated root). This is where the discriminant comes in handy. The discriminant, a part of the quadratic formula (b² - 4ac), tells us about the nature of the solutions without actually solving the equation. A positive discriminant means two real solutions, a zero discriminant means one real solution, and a negative discriminant means no real solutions.

Before we can analyze the discriminant, we need to rewrite the equation in the standard quadratic form: ax² + bx + c = 0. This involves moving all terms to one side of the equation, setting it equal to zero. Once we have it in this form, identifying the coefficients a, b, and c becomes straightforward, and we can plug them into the discriminant formula.

In our equation, x² + 8x = -20, we need to add 20 to both sides to get it into the standard form: x² + 8x + 20 = 0. Now we can clearly see that a = 1, b = 8, and c = 20. Let's calculate the discriminant: b² - 4ac = 8² - 4 * 1 * 20 = 64 - 80 = -16. A negative discriminant! This tells us that the equation has no real solutions, meaning the parabola does not intersect the x-axis.

So, armed with this knowledge, we can look at the graphs and eliminate any that show the parabola crossing the x-axis. The correct graph will be the one that shows a parabola that either hovers entirely above or entirely below the x-axis.

Calculating the Discriminant

In this section, we'll dive deeper into calculating the discriminant and understanding its significance in determining the nature of the solutions of a quadratic equation. As we discussed earlier, the discriminant is a crucial part of the quadratic formula, specifically the b² - 4ac portion, and it acts as a powerful tool for predicting the type and number of solutions an equation has without going through the entire process of solving it.

To calculate the discriminant, we first need to ensure our quadratic equation is in the standard form: ax² + bx + c = 0. This standard form is essential because it allows us to easily identify the coefficients a, b, and c, which are the key ingredients for our discriminant calculation. Once the equation is in the correct form, it's a simple matter of plugging these values into the discriminant formula.

Let's revisit our equation: x² + 8x = -20. To bring it to the standard form, we add 20 to both sides, resulting in x² + 8x + 20 = 0. Now, identifying the coefficients is straightforward: a = 1 (the coefficient of x²), b = 8 (the coefficient of x), and c = 20 (the constant term). These values are like the secret code to unlocking the nature of the solutions.

With the coefficients in hand, we can now calculate the discriminant: b² - 4ac. Plugging in our values, we get 8² - 4 * 1 * 20. Following the order of operations, we first calculate the square: 8² = 64. Then, we multiply: 4 * 1 * 20 = 80. Finally, we subtract: 64 - 80 = -16. So, our discriminant is -16.

But what does this negative value mean? This is where the interpretation of the discriminant comes into play. As a general rule:

• If the discriminant (b² - 4ac) is positive, the quadratic equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
• If the discriminant is zero, the quadratic equation has exactly one real solution (a repeated root). The parabola touches the x-axis at just one point, its vertex.
• If the discriminant is negative, the quadratic equation has no real solutions. This means the parabola does not intersect the x-axis at all; it hovers either entirely above or entirely below the x-axis.

In our case, the discriminant is -16, which is negative. Therefore, our equation x² + 8x + 20 = 0 has no real solutions. This crucial piece of information allows us to narrow down our choices when looking at the graphs. We can eliminate any graphs that show the parabola intersecting the x-axis because we know our equation doesn't have real roots.

Interpreting the Graph

Interpreting the graph of a quadratic equation is like reading a visual story of its solutions. The graph, a parabola, holds valuable information about the nature and number of solutions the equation possesses. Understanding how to extract this information is key to solving problems like the one we're tackling today.

The most important feature to look for when interpreting the graph is the intersection points of the parabola with the x-axis. These points, where the parabola crosses or touches the x-axis, represent the real solutions (or roots) of the quadratic equation. Remember, the solutions are the x-values that make the equation equal to zero.

If the parabola intersects the x-axis at two distinct points, it means the quadratic equation has two different real solutions. These solutions are simply the x-coordinates of the intersection points. Imagine the parabola cutting across the x-axis like a road crossing a river – two clear points of contact.

Now, if the parabola touches the x-axis at only one point, it indicates that the quadratic equation has exactly one real solution (a repeated root). In this scenario, the vertex of the parabola (the highest or lowest point) lies directly on the x-axis. It's like the parabola is gently kissing the x-axis at a single, special spot.

But what if the parabola doesn't intersect the x-axis at all? This is where things get interesting. When the parabola hovers entirely above or entirely below the x-axis, it means the quadratic equation has no real solutions. The solutions, in this case, are complex numbers, which are beyond the scope of what we can see on a standard Cartesian plane. Think of it as the parabola living in a different dimension, never quite touching our x-axis world.

Now, let's bring this back to our equation, x² + 8x + 20 = 0. We already calculated the discriminant and found it to be -16, which tells us that this equation has no real solutions. This means the graph representing this equation will be a parabola that does not intersect the x-axis. It will either be completely above the x-axis or completely below it.

So, when we look at the given graphs, we need to eliminate any options where the parabola crosses the x-axis. We're looking for a parabola that's floating either above or below, with no contact with the x-axis. This visual cue is the key to unlocking the correct answer. By understanding the relationship between the discriminant, the solutions, and the graph, we can confidently identify the graph that represents our equation.

Identifying the Correct Graph

Okay, guys, let's identify the correct graph that represents the solutions to our equation, x² + 8x = -20. We've done the groundwork, calculated the discriminant, and understood how to interpret graphs of quadratic equations. Now it's time to put it all together and nail this problem!

We know from our earlier calculations that the discriminant of the equation is -16. This crucial piece of information tells us that the equation has no real solutions. Remember, a negative discriminant means the parabola representing the equation does not intersect the x-axis.

So, what does this mean for us visually? It means we are looking for a graph where the parabola is either entirely above the x-axis or entirely below it. There should be no points where the parabola crosses or even touches the x-axis. It's like the parabola is playing a game of