Real Solutions Of Quadratic Equations: A Detailed Guide

Hey guys! Let's dive into the world of quadratic equations and figure out how to determine the number of real solutions they possess. We'll take a specific example, the equation 2x + 16 = 4x² + 4, and break it down step by step. Understanding this concept is crucial, not just for math class, but also for various real-world applications where quadratic equations pop up. So, buckle up, and let's get started!

Understanding the Quadratic Equation

To really nail down how many real solutions our equation has, we need to first transform it into the standard quadratic form. This form is expressed as ax² + bx + c = 0. Why this form? Because it's super handy for applying the tools and techniques we have for solving quadratics. Let's get that equation massaged into shape! So in this paragraph, you'll understand the standard quadratic form and how it helps us find the solutions. Stick around as we dive deep into this topic. Remember, mastering this skill opens doors to tackling many mathematical challenges.

First, let’s rearrange the given equation: 2x + 16 = 4x² + 4. Our goal is to set one side of the equation to zero. We can achieve this by moving all terms to the right side. This gives us: 0 = 4x² - 2x + 4 - 16, which simplifies to 0 = 4x² - 2x - 12. Now, we have a quadratic equation in the standard form: 4x² - 2x - 12 = 0. Identifying the coefficients is the next crucial step. In our equation, 'a' (the coefficient of x²) is 4, 'b' (the coefficient of x) is -2, and 'c' (the constant term) is -12. These coefficients are the keys to unlocking the secrets of the quadratic equation, specifically the number of real solutions.

By having the equation in standard form and knowing the coefficients, we can now use the discriminant. The discriminant, denoted as Δ (Delta), is a critical component of the quadratic formula, which is Δ = b² - 4ac. This simple expression holds immense power, as its value directly tells us about the nature and number of solutions. Before we even try to solve for x, the discriminant gives us a sneak peek into what kind of answers to expect. It tells us if we'll find real, distinct solutions, a single real solution, or no real solutions at all. So, understanding the standard form and the discriminant is like having a secret decoder for quadratic equations – it makes the whole solving process much clearer and more efficient!

The Discriminant: Your Key to Solutions

Let's talk about the discriminant, which is the real MVP when determining the nature of solutions in a quadratic equation. The discriminant, represented as Δ (Delta), is calculated using the formula Δ = b² - 4ac. This little formula is super powerful because it tells us how many real solutions our quadratic equation has without us even having to fully solve it! The discriminant really helps simplify the whole process of figuring out what kind of solutions to expect.

So, how does the discriminant actually work? Well, it all comes down to its value. If Δ is greater than zero (Δ > 0), then the quadratic equation has two distinct real solutions. This means there are two different values of x that will make the equation true. Think of it like two separate paths leading to the solution. When Δ is equal to zero (Δ = 0), the equation has exactly one real solution (or, you could say, two equal real solutions). In this case, there's only one value of x that satisfies the equation. It's like the two paths have merged into one. Lastly, if Δ is less than zero (Δ < 0), the equation has no real solutions. This means there are no values of x on the real number line that will make the equation true. The solutions, in this case, are complex numbers, which are a topic for another day. So, in essence, the discriminant is like a traffic light for solutions: green for two, yellow for one, and red for none!

Now, let’s apply this to our equation: 4x² - 2x - 12 = 0. Remember, a = 4, b = -2, and c = -12. Plugging these values into the discriminant formula, we get: Δ = (-2)² - 4 * 4 * (-12). This simplifies to Δ = 4 + 192, which gives us Δ = 196. Since 196 is clearly greater than zero, we know that our equation has two distinct real solutions. Isn't that neat? Just by calculating one value, we've already determined the number of real solutions. This is why the discriminant is such a valuable tool in the world of quadratic equations. It saves us time and gives us a clear understanding of what to expect when we dive into solving for x.

Calculating the Discriminant for 2x + 16 = 4x² + 4

Alright, let's get our hands dirty and actually calculate the discriminant for our specific equation: 2x + 16 = 4x² + 4. Remember, the first step is to get the equation into that sweet, sweet standard form: ax² + bx + c = 0. We already did this earlier, but let's quickly recap to keep things crystal clear. We rearranged the equation to get 4x² - 2x - 12 = 0. Now, identifying our coefficients is super easy: a = 4, b = -2, and c = -12. These are the magic numbers we'll be plugging into the discriminant formula.

The discriminant formula, as we know, is Δ = b² - 4ac. It's a good idea to memorize this formula, guys, as it's going to be your best friend when dealing with quadratic equations. Now, let’s substitute our values: Δ = (-2)² - 4 * 4 * (-12). See how we're just replacing the letters with the corresponding numbers? Simple as pie! Next, we need to do the math. First, (-2)² is 4. Then, 4 * 4 * (-12) is -192. So, our equation now looks like this: Δ = 4 - (-192). Remember, subtracting a negative is the same as adding, so we get Δ = 4 + 192. Finally, adding those together gives us Δ = 196.

So, we've done it! We've calculated the discriminant for our equation, and we found that Δ = 196. This is a positive number, which, as we discussed earlier, means our quadratic equation has two distinct real solutions. High five! We’ve successfully navigated the calculation and interpreted the result. This step-by-step approach is key to mastering these concepts. By carefully plugging in the values and doing the arithmetic, we've unlocked a crucial piece of information about our equation. This process not only gives us the answer but also builds our confidence in tackling more complex problems in the future. Keep practicing, and you'll become a discriminant-calculating pro in no time!

Determining the Number of Real Solutions

Now that we've calculated the discriminant (Δ = 196) for our equation 2x + 16 = 4x² + 4, it's time to put that knowledge to use and determine the number of real solutions. Remember, the discriminant is like our guide, telling us exactly what to expect in terms of solutions. We know that a positive discriminant means there are two distinct real solutions, a discriminant of zero means there is one real solution, and a negative discriminant means there are no real solutions. So, let’s see what our Δ = 196 tells us.

Since our discriminant, 196, is greater than zero, we can confidently say that the equation 4x² - 2x - 12 = 0 has two distinct real solutions. This means there are two different values of x that will satisfy the equation. We don't even need to solve for those values yet; we already know they exist! That’s the power of the discriminant – it gives us a quick and easy way to understand the nature of the solutions. This is super helpful because it saves us time and effort. Imagine trying to solve a quadratic equation only to find out later that there are no real solutions – the discriminant helps us avoid that!

To further solidify this concept, let's think about what this means graphically. A quadratic equation, when graphed, forms a parabola. The real solutions of the equation correspond to the points where the parabola intersects the x-axis. So, if there are two real solutions, the parabola crosses the x-axis at two different points. If there is one real solution, the parabola touches the x-axis at exactly one point (the vertex). And if there are no real solutions, the parabola doesn't intersect the x-axis at all. In our case, since we have two real solutions, the parabola will cross the x-axis at two points. Understanding this visual representation can be incredibly helpful in grasping the concept of real solutions. So, by calculating the discriminant and understanding its implications, we've successfully determined that our equation has two real solutions. This is a significant step in mastering quadratic equations!

Justifying the Answer

To wrap things up, let's formally justify our answer regarding the number of real solutions for the equation 2x + 16 = 4x² + 4. Justification is super important in mathematics, guys. It’s not enough to just get the right answer; we need to be able to explain why our answer is correct. This shows a deeper understanding of the concepts involved and helps us build a solid foundation for more advanced topics. So, let’s put on our explanation hats and walk through the logical steps we took to arrive at our conclusion.

Our justification starts with transforming the given equation into the standard quadratic form: ax² + bx + c = 0. We did this by rearranging the terms in 2x + 16 = 4x² + 4 to get 4x² - 2x - 12 = 0. This step is crucial because it allows us to easily identify the coefficients a, b, and c, which are essential for calculating the discriminant. Next, we identified the coefficients: a = 4, b = -2, and c = -12. These values are the building blocks for the next step, so it's vital to get them right.

Then, we calculated the discriminant using the formula Δ = b² - 4ac. Plugging in our values, we got Δ = (-2)² - 4 * 4 * (-12), which simplified to Δ = 196. This is where the magic happens! The discriminant is the key to unlocking the number of real solutions. Since our discriminant is positive (Δ = 196 > 0), we know that the quadratic equation has two distinct real solutions. This is the core of our justification: the positive discriminant definitively tells us that there are two real solutions. We can confidently say that the correct answer is (c) Two, because our calculated discriminant is positive, proving that the equation has two real solutions. By outlining these steps, we've not only answered the question but also demonstrated a clear understanding of the underlying principles. This is what true mathematical understanding is all about!

In conclusion, by transforming the equation to standard form, calculating the discriminant, and interpreting its value, we have confidently determined that the equation 2x + 16 = 4x² + 4 has two real solutions. Remember, the discriminant is your friend when it comes to quadratic equations! Keep practicing, and you'll master these concepts in no time.