Solving Quadratic Equations Using The Quadratic Formula

Hey everyone! Today, we're going to dive into solving a quadratic equation using the quadratic formula. Quadratic equations might seem intimidating at first, but don't worry, we'll break it down step by step so you can master them. We'll tackle the equation $4x^2 - 3x + 9 = 2x + 1$ and find the values of $x$ that satisfy it. Let's get started!

Understanding Quadratic Equations and the Quadratic Formula

Before we jump into solving our specific equation, let's make sure we're all on the same page about quadratic equations and the quadratic formula. So, what exactly is a quadratic equation? Simply put, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x) is 2. The general form of a quadratic equation is given by: $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero (because if a were zero, it wouldn't be a quadratic equation anymore, right?). These constants are super important, as we'll see when we use the quadratic formula. The solutions to a quadratic equation are also known as its roots or zeros, and these are the values of x that make the equation true. A quadratic equation can have two real solutions, one real solution (which we call a repeated root), or two complex solutions. Complex solutions involve imaginary numbers, which we'll touch on later when we encounter a negative number inside a square root. Now, where does the quadratic formula come in? Well, the quadratic formula is a powerful tool that provides a direct way to find the solutions of any quadratic equation, no matter how messy it looks. It's derived by completing the square on the general form of the quadratic equation, and it's a formula you'll definitely want to have in your math toolkit. The quadratic formula is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Notice those constants a, b, and c we talked about? They're the stars of the show here! To use the formula, you just need to identify the values of a, b, and c from your quadratic equation, plug them into the formula, and simplify. The "$\pm$" symbol means "plus or minus," indicating that there are potentially two solutions: one where you add the square root part and one where you subtract it. The expression inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If it's zero, we have one real solution (a repeated root). And if it's negative, we have two complex solutions. So, the discriminant is like a little detective that gives us clues about the solutions before we even fully solve the equation. Understanding the quadratic formula and what each part represents is crucial for successfully solving quadratic equations. It might seem like a lot to remember at first, but with practice, it'll become second nature. Now that we've got a good grasp of the basics, let's apply this knowledge to our specific problem.

Transforming the Equation into Standard Form

The first crucial step in solving any quadratic equation using the quadratic formula is to make sure our equation is in the standard form: $ax^2 + bx + c = 0$. Currently, our equation is given as $4x^2 - 3x + 9 = 2x + 1$. As you can see, it's not quite in the standard form yet because we have terms on both sides of the equation. To get it into the standard form, we need to move all the terms to one side, leaving zero on the other side. We can achieve this by subtracting $2x$ and $1$ from both sides of the equation. This ensures that we maintain the equality while rearranging the terms. So, let's do it: $4x^2 - 3x + 9 - 2x - 1 = 2x + 1 - 2x - 1$. Now, we simplify the equation by combining like terms. On the left side, we have $-3x$ and $-2x$, which combine to give us $-5x$. We also have $9$ and $-1$, which combine to give us $8$. On the right side, $2x$ and $-2x$ cancel out, and $1$ and $-1$ also cancel out, leaving us with zero. This gives us the transformed equation: $4x^2 - 5x + 8 = 0$. Now, this is the standard form of a quadratic equation, and it's perfect for applying the quadratic formula. Notice that we have a term with $x^2$, a term with $x$, and a constant term, all set equal to zero. This makes it easy to identify the coefficients a, b, and c, which are the key ingredients for the quadratic formula. Transforming the equation into standard form is a fundamental step because the quadratic formula is specifically designed to work with equations in this form. If we try to apply the formula before rearranging the equation, we'll end up with incorrect values for a, b, and c, and consequently, incorrect solutions. So, always remember to get the equation into the form $ax^2 + bx + c = 0$ first. It's like laying the foundation for a building – you can't build a sturdy structure without a solid base. Once the equation is in standard form, we can confidently move on to the next step: identifying the coefficients and plugging them into the quadratic formula. This is where the real magic happens, and we start to see the solutions emerge. So, with our equation now in the perfect format, let's proceed to the next step and unlock the solutions.

Identifying Coefficients a, b, and c

Now that we have our quadratic equation in the standard form, $4x^2 - 5x + 8 = 0$, the next step is to carefully identify the coefficients a, b, and c. These coefficients are the numerical values that multiply the terms $x^2$, $x$, and the constant term, respectively. Remember, these values are crucial because they are the key inputs for the quadratic formula. So, let's break down our equation and pinpoint these coefficients. The coefficient a is the number that multiplies the $x^2$ term. In our equation, the term with $x^2$ is $4x^2$, so the coefficient a is 4. It's important to pay attention to the sign of the coefficient. In this case, a is positive, but if the term were $-4x^2$, then a would be -4. Next, we identify the coefficient b, which is the number that multiplies the x term. In our equation, the x term is $-5x$, so the coefficient b is -5. Again, the sign is very important here. If we were to miss the negative sign and use 5 instead of -5, it would lead to an incorrect solution. Finally, we identify the coefficient c, which is the constant term. In our equation, the constant term is 8, so the coefficient c is 8. It's the term that doesn't have any x attached to it. So, to summarize, in the equation $4x^2 - 5x + 8 = 0$, we have identified the coefficients as follows: a = 4, b = -5, and c = 8. Correctly identifying these coefficients is a fundamental step because these values will be plugged directly into the quadratic formula. If we make a mistake in this step, the rest of the solution will be incorrect. It's like entering the wrong numbers into a calculator – you'll get the wrong result, no matter how carefully you perform the calculation. Therefore, it's always a good idea to double-check your values for a, b, and c before proceeding to the next step. With the coefficients now correctly identified, we're ready to plug them into the quadratic formula and solve for x. This is where the magic of the formula comes into play, and we'll see how it helps us find the solutions to our quadratic equation. So, let's move on to the exciting part – applying the quadratic formula!

Applying the Quadratic Formula

Alright, we've got our quadratic equation in standard form, $4x^2 - 5x + 8 = 0$, and we've carefully identified our coefficients: a = 4, b = -5, and c = 8. Now comes the exciting part – plugging these values into the quadratic formula! Remember the formula? It's: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula might look a little intimidating, but don't worry, we'll take it one step at a time. The key is to substitute the values of a, b, and c carefully and then simplify. So, let's start by plugging in our values. We have b = -5, so -b becomes -(-5), which is just 5. Then we have the $\pm$ symbol, followed by the square root. Inside the square root, we have $b^2$, which is (-5)^2, which equals 25. Next, we have -4ac, which is -4 * 4 * 8. Multiplying those together gives us -128. So, inside the square root, we have 25 - 128. In the denominator, we have 2a, which is 2 * 4, which equals 8. Putting it all together, our equation now looks like this: $x = \frac{5 \pm \sqrt{25 - 128}}{8}$. Now, let's simplify what's under the square root. We have 25 - 128, which equals -103. So, our equation becomes: $x = \frac{5 \pm \sqrt{-103}}{8}$. Notice that we have a negative number inside the square root. This tells us that we're going to have complex solutions, which involve imaginary numbers. Remember that the square root of -1 is defined as the imaginary unit, i. So, we can rewrite $\sqrt{-103}$ as $\sqrt{103} * \sqrt{-1}$, which is $\sqrt{103}i$. Now, our equation looks like this: $x = \frac{5 \pm \sqrt{103}i}{8}$. And there you have it! We've applied the quadratic formula and simplified the expression as much as possible. The solutions for x are $\frac{5 + \sqrt{103}i}{8}$ and $\frac{5 - \sqrt{103}i}{8}$. These are complex solutions because they involve the imaginary unit i. Applying the quadratic formula can seem tricky at first, but with practice, it becomes much easier. The key is to be careful with your substitutions and to follow the order of operations. And remember, if you encounter a negative number under the square root, it just means you have complex solutions. Now that we've found the solutions, let's take a look at how they relate to the options provided in the question.

Matching the Solution with the Options

Okay, we've successfully navigated the quadratic formula and arrived at our solutions: $x = \frac{5 \pm \sqrt{103}i}{8}$. Now, the final step is to compare these solutions with the options provided in the original question and identify the correct answer. Let's recap the options:

A. $\frac{1 \pm \sqrt{159} i}{8}$ B. $\frac{5 \pm \sqrt{153} i}{8}$ C. $\frac{5 \pm \sqrt{103} i}{8}$ D. $\frac{1 \pm \sqrt{153}}{8}$

By carefully comparing our solutions with the options, we can see that our solutions, $\frac{5 \pm \sqrt{103}i}{8}$, exactly match option C. Option A has a different numerator (1 instead of 5) and a different number under the square root (159 instead of 103). Option B has the correct numerator (5), but a different number under the square root (153 instead of 103). Option D is also incorrect because it has a different numerator (1), a different number under the square root (153), and it doesn't include the imaginary unit i, indicating that it's a real solution, not a complex one like ours. Therefore, by process of elimination and direct comparison, we can confidently conclude that the correct answer is option C. Matching the solution with the options is a crucial step because it ensures that we haven't made any mistakes along the way. It's like the final checkmark on a to-do list – it confirms that we've successfully completed the task. In this case, it validates that our calculations and simplifications were accurate, and we've arrived at the correct solution. So, always take the time to compare your solution with the provided options, especially in multiple-choice questions. It's a simple step that can prevent careless errors and ensure that you get the right answer. And with that, we've successfully solved the quadratic equation using the quadratic formula and identified the correct answer! Great job, guys!

Conclusion: Mastering the Quadratic Formula

Woo-hoo! We did it! We successfully used the quadratic formula to solve the equation $4x^2 - 3x + 9 = 2x + 1$, and we found that the values of x are $\frac{5 \pm \sqrt{103}i}{8}$, which corresponds to option C. This journey through the quadratic formula might have seemed a bit challenging at first, but hopefully, you now feel more confident in your ability to tackle these types of problems. Solving quadratic equations is a fundamental skill in algebra, and the quadratic formula is a powerful tool that can help you find solutions even when other methods, like factoring, don't work. We started by understanding what a quadratic equation is and recognizing its standard form, $ax^2 + bx + c = 0$. Then, we dove into the quadratic formula itself, learning how it's derived and what each part represents. We emphasized the importance of correctly identifying the coefficients a, b, and c because these values are the key inputs for the formula. We then meticulously plugged these values into the formula, simplified the expression, and dealt with the imaginary unit i when we encountered a negative number under the square root. Finally, we compared our solutions with the given options and confidently selected the correct answer. The key takeaway here is that the quadratic formula is a reliable method for solving any quadratic equation, but it requires careful attention to detail and a systematic approach. Don't be afraid to break down the problem into smaller steps, and always double-check your work to avoid careless errors. Remember, practice makes perfect! The more you use the quadratic formula, the more comfortable and confident you'll become with it. Try solving different quadratic equations with varying coefficients and see how the formula works in different scenarios. You can also explore the relationship between the discriminant ($b^2 - 4ac$) and the nature of the solutions (real or complex). So, keep practicing, keep exploring, and keep building your math skills. The quadratic formula is just one tool in your mathematical toolbox, but it's a powerful one that can open doors to more advanced concepts. And remember, math can be challenging, but it's also incredibly rewarding. The feeling of solving a tough problem and understanding a new concept is truly satisfying. So, embrace the challenge, keep learning, and never stop asking questions. You've got this!