Solving $3x^2 + 6x + 10 = 0$ With The Quadratic Formula A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of quadratic equations, and more specifically, how to solve them using the quadratic formula. If you've ever stared blankly at an equation like $3x^2 + 6x + 10 = 0$ and wondered where to even begin, you're in the right place. This guide will break down the quadratic formula step-by-step, making it super easy to understand and use. We'll use the example equation $3x^2 + 6x + 10 = 0$ to illustrate each step, so you can follow along and see exactly how it works. No more quadratic equation panic – let's get started!

Understanding Quadratic Equations

Before we jump into the formula itself, let's make sure we're all on the same page about what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. What does that mean? Well, it means it can be written in the general form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and a is not equal to zero. The x is our variable, the thing we're trying to solve for. The coefficients a, b, and c play crucial roles in determining the solutions (also called roots) of the equation. Now, why do we care about quadratic equations? They pop up everywhere in math and real-world applications! From physics problems involving projectile motion to engineering calculations and even financial modeling, quadratic equations are essential tools. Recognizing them and knowing how to solve them is a key skill in many fields. Think about throwing a ball – the path it follows can be described by a quadratic equation. Or consider designing a bridge – engineers use quadratic equations to ensure stability and safety. Even in finance, calculating compound interest involves quadratic relationships. So, understanding these equations opens up a whole world of problem-solving possibilities. But sometimes, solving these equations isn't as straightforward as simple algebra. That's where the quadratic formula comes to the rescue!

Identifying Coefficients

The first step in using the quadratic formula is to correctly identify the coefficients a, b, and c in your equation. This might seem simple, but it's super important to get right because a small mistake here can throw off your entire solution. Remember, the general form of a quadratic equation is $ax^2 + bx + c = 0$. So, a is the coefficient of the $x^2$ term, b is the coefficient of the x term, and c is the constant term. Let's practice with our example equation: $3x^2 + 6x + 10 = 0$. In this case:

• a = 3 (the coefficient of $x^2$)
• b = 6 (the coefficient of x)
• c = 10 (the constant term)

See? Not too tricky! But let's look at a few more examples to make sure we've got it down. What if our equation was $x^2 - 5x + 4 = 0$? Here, a would be 1 (since $x^2$ is the same as $1x^2$), b would be -5 (don't forget the negative sign!), and c would be 4. How about an equation like $2x^2 + 7 = 0$? In this case, a is 2, b is 0 (since there's no x term), and c is 7. Getting comfortable with identifying a, b, and c is like laying the foundation for a building – if it's solid, everything else will stand strong. So, take your time, double-check your work, and you'll be solving quadratic equations like a pro in no time!

The Quadratic Formula: Your New Best Friend

Okay, now for the star of the show: the quadratic formula. This formula is a magical tool that allows you to find the solutions (or roots) of any quadratic equation, no matter how complicated it looks. It might seem a bit intimidating at first glance, but trust me, once you break it down, it's totally manageable. Here it is:

x = rac{-b old{\pm} old{\sqrt{b^2 - 4ac}}}{2a}

Whoa! Lots of symbols, right? Don't worry, we'll go through it piece by piece. The x on the left side represents the solutions we're trying to find. The ± symbol means we'll actually get two solutions – one where we add the square root part and one where we subtract it. The rest of the letters – a, b, and c – are the coefficients we just learned how to identify! The formula essentially takes these coefficients, plugs them into a specific arrangement, and spits out the values of x that make the equation true. It's like a secret decoder ring for quadratic equations! So, where does this formula come from? Well, it's derived by a process called "completing the square," which is a technique for rewriting quadratic equations in a more convenient form. The quadratic formula is the result of applying completing the square to the general form of the quadratic equation ($ax^2 + bx + c = 0$). While the derivation is interesting, the most important thing for now is to understand how to use the formula. Think of it as a recipe – you don't need to know the chemistry behind baking a cake to follow the instructions and get a delicious result. Similarly, you don't need to know the derivation of the quadratic formula to use it effectively. Just plug in the coefficients, follow the steps, and you'll be solving equations left and right!

Breaking Down the Formula

Let's take a closer look at each part of the quadratic formula to really understand what's going on. This will make it much easier to use and remember. The first part we encounter is “-b”. This simply means we take the value of b (the coefficient of the x term) and change its sign. If b is positive, we make it negative, and if b is negative, we make it positive. It's a straightforward step, but it's crucial to get it right. Next up, we have the “±” symbol. This is super important because it tells us that we're actually going to get two solutions for x. One solution comes from adding the square root part, and the other comes from subtracting it. Quadratic equations often have two solutions because of the squared term ($x^2$). Now, let's dive into the heart of the formula: the square root part, “√(b² - 4ac)”. This part is called the discriminant, and it tells us a lot about the nature of the solutions. We'll talk more about the discriminant later, but for now, let's focus on calculating it. Inside the square root, we first square b (multiply b by itself). Then, we subtract 4 times a times c. Remember to follow the order of operations (PEMDAS/BODMAS) when calculating this part! Finally, we divide the entire expression by “2a”, which means we multiply the coefficient a by 2. So, to recap, the quadratic formula is like a mini-program that takes a, b, and c as inputs and gives us the solutions for x. By breaking it down into smaller parts, we can see that each step is manageable and logical. Don't be intimidated by the formula – embrace it as a powerful tool in your mathematical arsenal!

Applying the Quadratic Formula to $3x^2 + 6x + 10 = 0$

Alright, let's put our newfound knowledge to the test! We're going to use the quadratic formula to solve our example equation: $3x^2 + 6x + 10 = 0$. Remember, the first step is to identify the coefficients a, b, and c. We already did this earlier, but let's recap:

• a = 3
• b = 6
• c = 10

Now comes the fun part: plugging these values into the quadratic formula:

x = rac{-b old{\pm} old{\sqrt{b^2 - 4ac}}}{2a}

Substitute the values:

x = rac{-6 old{\pm} old{\sqrt{6^2 - 4(3)(10)}}}{2(3)}

See how we just replaced a, b, and c with their corresponding numbers? Now, we need to simplify this expression. Let's start with the part under the square root:

$6^2 - 4(3)(10) = 36 - 120 = -84$

Uh oh! We have a negative number under the square root. This is a clue that our solutions will be complex numbers (numbers involving the imaginary unit i, where i = old{\sqrt{-1}}). Don't panic! We can still handle this. Let's continue substituting this back into the formula:

x = rac{-6 old{\pm} old{\sqrt{-84}}}{6}

Now, we need to simplify the square root of -84. We can rewrite old{\sqrt{-84}} as old{\sqrt{84}} old{\cdot} old{\sqrt{-1}}. We know that old{\sqrt{-1}} is i, and we can simplify old{\sqrt{84}} by finding its prime factors. 84 can be factored as 2 old{\cdot} 2 old{\cdot} 3 old{\cdot} 7, so old{\sqrt{84}} = old{\sqrt{2^2 old{\cdot} 3 old{\cdot} 7}} = 2old{\sqrt{21}}. So, old{\sqrt{-84}} = 2old{\sqrt{21}}i. Let's substitute this back into the equation:

x = rac{-6 old{\pm} 2old{\sqrt{21}}i}{6}

Finally, we can simplify this expression by dividing both the real and imaginary parts by 6:

x = -1 old{\pm} rac{old{\sqrt{21}}}{3}i

So, our solutions are:

• x_1 = -1 + rac{old{\sqrt{21}}}{3}i
• x_2 = -1 - rac{old{\sqrt{21}}}{3}i

These are complex conjugate solutions. Congratulations! You've just solved a quadratic equation using the quadratic formula and encountered complex numbers along the way. You're becoming a quadratic equation master!

Understanding the Discriminant

We briefly mentioned the discriminant earlier, but let's dive deeper into why it's so important. The discriminant is the part of the quadratic formula under the square root: $b^2 - 4ac$. This little expression holds the key to understanding the nature of the solutions of a quadratic equation. It tells us whether the solutions are real or complex, and how many solutions there are. There are three main scenarios:

1. If $b^2 - 4ac > 0$ (positive): The equation has two distinct real solutions. This means the graph of the quadratic equation (a parabola) intersects the x-axis at two different points. Think of it like this: the positive discriminant gives the quadratic formula the "go-ahead" to find two different real roots. The square root of a positive number is a real number, so the ± in the formula leads to two distinct values for x.
2. If $b^2 - 4ac = 0$ (zero): The equation has exactly one real solution (a repeated root). This means the parabola touches the x-axis at exactly one point. When the discriminant is zero, the square root part of the quadratic formula disappears. This leaves us with only one solution, x = -b / 2a. This single solution represents the point where the parabola just kisses the x-axis.
3. If $b^2 - 4ac < 0$ (negative): The equation has two complex solutions (conjugate pairs). This means the parabola does not intersect the x-axis. As we saw in our example problem, a negative discriminant leads to taking the square root of a negative number, which results in imaginary numbers. Complex solutions always come in conjugate pairs, meaning they have the form a + bi and a - bi, where i is the imaginary unit (old{\sqrt{-1}}).

So, by simply calculating the discriminant, we can get a sneak peek at the type of solutions we can expect. It's like a weather forecast for quadratic equations – it tells us what kind of mathematical landscape we're about to encounter!

Calculating the Discriminant for $3x^2 + 6x + 10 = 0$

Let's calculate the discriminant for our example equation, $3x^2 + 6x + 10 = 0$, to confirm what we already observed. We have a = 3, b = 6, and c = 10. The discriminant is:

$b^2 - 4ac = 6^2 - 4(3)(10) = 36 - 120 = -84$

As we found earlier, the discriminant is -84, which is negative. This confirms that our equation has two complex solutions, just like we saw when we solved it using the quadratic formula. Calculating the discriminant beforehand can save you time and effort. If you find that the discriminant is negative, you know you're dealing with complex solutions and can prepare for the imaginary numbers that will pop up. If it's positive, you know you'll have two real solutions, and if it's zero, you'll have one real solution. It's a handy little shortcut to add to your problem-solving toolkit!

Conclusion: Mastering the Quadratic Formula

Wow, we've covered a lot in this guide! You've learned what quadratic equations are, how to identify their coefficients, how to use the quadratic formula, and how to interpret the discriminant. You even tackled an equation with complex solutions! By now, you should feel much more confident in your ability to solve quadratic equations. The quadratic formula is a powerful tool, and with practice, you'll become a pro at using it. Remember, the key is to break down the problem into smaller steps: identify a, b, and c, plug the values into the formula, simplify carefully, and don't forget to consider the discriminant. Keep practicing, and you'll be solving quadratic equations in your sleep (well, maybe not, but you'll definitely be good at it!). So, the next time you encounter a quadratic equation, don't shy away. Embrace the challenge, whip out the quadratic formula, and show that equation who's boss! You've got this!