Solving Systems Of Equations Determining Real Solutions

by Axel Sørensen 56 views

Hey everyone! Let's dive into a fascinating math problem together. We're going to explore a system of equations and figure out just how many real solutions it has. Get ready to put on your thinking caps, because this is going to be an exciting journey into the world of algebra!

The Challenge: Deciphering the System of Equations

Our mission, should we choose to accept it (and we totally do!), is to determine the number of real solutions for this system of equations:

y = x^2 + x + 3
y = -2x - 5

Before we jump into solving, let's take a moment to appreciate what we're looking at. We have two equations, each representing a different kind of curve on a graph. The first equation, y = x^2 + x + 3, is a quadratic equation, which means it graphs as a parabola – a U-shaped curve. The second equation, y = -2x - 5, is a linear equation, which means it graphs as a straight line.

The solutions to this system of equations are the points where these two graphs intersect. Think of it like a road map – the solutions are the places where the parabola and the line cross paths. Our goal is to figure out how many of these intersections exist, which will tell us how many real solutions we have. So, buckle up, because we're about to embark on a quest to find those intersections!

Why are we even doing this? Well, understanding systems of equations is crucial in many areas of math and science. They help us model real-world situations, from the trajectory of a ball to the optimal pricing strategy for a business. By mastering these concepts, we're not just solving equations; we're building the skills to tackle complex problems in all sorts of fields.

Now, let's get down to the nitty-gritty of actually solving this problem. We'll explore different methods and strategies, making sure to break down each step so it's crystal clear. So, are you ready to become a solution-sleuthing superhero? Let's go!

Method 1: The Power of Substitution

Our first approach to unraveling this system of equations involves a clever technique called substitution. This method is like a mathematical magic trick – we're going to swap one expression for another to simplify our problem. The core idea is to use one equation to express one variable in terms of the other, and then plug that expression into the second equation.

In our case, we have:

y = x^2 + x + 3
y = -2x - 5

Notice that both equations are already solved for y. This is fantastic news! It means we can directly substitute the expression for y from the second equation into the first equation. We're essentially saying, "Hey, if y is equal to both of these things, then those things must be equal to each other!"

So, let's perform the substitution:

x^2 + x + 3 = -2x - 5

Ta-da! We've transformed our system of two equations into a single equation with just one variable, x. This is a huge step forward. Now, our goal is to solve this equation for x. To do that, we'll need to rearrange the equation into a standard form that we can work with. The form we're aiming for is the quadratic form: ax^2 + bx + c = 0, where a, b, and c are constants.

To get there, let's move all the terms to one side of the equation. We'll add 2x and 5 to both sides:

x^2 + x + 3 + 2x + 5 = 0

Now, let's combine like terms to simplify:

x^2 + 3x + 8 = 0

Awesome! We've successfully transformed our equation into the standard quadratic form. Now, we have a powerful arsenal of tools at our disposal to solve for x. We could try factoring, completing the square, or using the quadratic formula. But before we dive into those techniques, let's take a step back and think strategically.

Our original question wasn't to find the exact values of x, but rather to determine how many real solutions exist. This hints that there might be a more efficient way to answer the question without fully solving for x. We'll explore that in the next section. So, stay tuned, because we're about to unlock a shortcut that will save us time and effort!

Method 2: The Discriminant Detective

Remember that quadratic equation we arrived at in the previous section, x^2 + 3x + 8 = 0? Well, it holds the key to unlocking the number of real solutions without actually solving for x. This is where the discriminant comes into play – a mathematical detective that helps us determine the nature of the roots of a quadratic equation.

The discriminant is a special part of the quadratic formula, which is used to solve quadratic equations of the form ax^2 + bx + c = 0. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant is the expression under the square root: b^2 - 4ac. This little expression packs a powerful punch. Its value tells us everything we need to know about the number and type of solutions to the quadratic equation.

Here's how the discriminant works its magic:

  • If b^2 - 4ac > 0 (positive), the equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
  • If b^2 - 4ac = 0 (zero), the equation has one real solution (a repeated root). This means the parabola touches the x-axis at exactly one point.
  • If b^2 - 4ac < 0 (negative), the equation has no real solutions. This means the parabola doesn't intersect the x-axis at all.

So, the discriminant acts like a crystal ball, revealing the secrets of our quadratic equation without us having to do all the heavy lifting of solving it completely. Now, let's put this detective work into action!

In our equation, x^2 + 3x + 8 = 0, we have:

  • a = 1 (the coefficient of x^2)
  • b = 3 (the coefficient of x)
  • c = 8 (the constant term)

Let's plug these values into the discriminant formula:

b^2 - 4ac = (3)^2 - 4(1)(8) = 9 - 32 = -23

Aha! The discriminant is -23, which is a negative number. According to our discriminant detective's wisdom, this means our quadratic equation has no real solutions. That's a pretty neat trick, right? We've determined the number of solutions without even having to find them!

But what does this mean in the context of our original system of equations? Remember, the solutions to the system are the points where the parabola and the line intersect. Since our quadratic equation has no real solutions, it means the parabola and the line never cross paths. They exist in the same mathematical universe, but they never touch. It's like two ships passing in the night, or maybe two parallel lines that never meet. Whatever analogy you prefer, the bottom line is: there are no intersection points, and therefore, no real solutions to the system of equations.

The Grand Finale: Choosing the Correct Answer

We've journeyed through the world of systems of equations, wielding the power of substitution and the wisdom of the discriminant. We've transformed equations, calculated values, and interpreted results. Now, it's time to reap the rewards of our hard work and choose the correct answer to our original question: How many solutions does this system of equations have?

y = x^2 + x + 3
y = -2x - 5

We've discovered that the discriminant of the resulting quadratic equation is negative, which means there are no real solutions. This corresponds to option A in the original question.

So, the final answer is:

A. no real solutions

Congratulations, everyone! We've successfully navigated this mathematical challenge. We've not only found the answer but also gained a deeper understanding of systems of equations, quadratic equations, and the power of the discriminant. We've learned how to approach problems strategically, using the right tools and techniques to arrive at the solution efficiently.

But the journey doesn't end here. There's always more to explore in the fascinating world of mathematics. So, keep asking questions, keep experimenting, and keep pushing your mathematical boundaries. Who knows what amazing discoveries you'll make along the way? Now, go forth and conquer more mathematical mountains!