Solving 4x^2 + 3x - 5 = 0 Which Expression To Use

Aug 7, 2025 by ADMIN 50 views

Solving Quadratic Equations Demystified Answering the Question Which Expression Could You Use to Solve 4x^2 + 3x - 5 = 0

Hey guys! Today, let's dive into the fascinating world of quadratic equations. Specifically, we're going to tackle the question: Which expression could you use to solve the quadratic equation $4x^2 + 3x - 5 = 0$? Quadratic equations might seem intimidating at first, but trust me, once you understand the basics and the powerful tool we have – the quadratic formula – they become much less scary. So, buckle up, and let's get started!

Understanding Quadratic Equations

Before we jump into solving the equation, 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 the highest power of the variable (usually 'x') in the equation is 2. The standard form of a quadratic equation is expressed as: $ax^2 + bx + c = 0$ where a, b, and c are constants, and 'a' is not equal to zero (because if 'a' were zero, the equation would become linear, not quadratic). In our specific equation, $4x^2 + 3x - 5 = 0$, we can easily identify the coefficients: a = 4, b = 3, and c = -5. These coefficients are super important because they're the keys to unlocking the solutions using the quadratic formula.

Now, why do we care about solving quadratic equations? These equations pop up everywhere in the real world! From physics problems involving projectile motion to engineering designs and even financial modeling, quadratic equations are essential tools for describing and understanding various phenomena. Learning how to solve them opens up a whole new world of problem-solving capabilities. Think about calculating the trajectory of a ball thrown in the air, determining the dimensions of a garden to maximize its area, or even predicting the growth of a population. All of these scenarios can involve quadratic equations. So, mastering this concept is definitely worth the effort!

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Factoring is a great method when the equation can be easily factored, but it's not always straightforward. Completing the square is another powerful technique, but it can be a bit more involved. The quadratic formula, on the other hand, is a universal tool that works for any quadratic equation, no matter how complex. That's why it's often the go-to method for solving these types of problems. In this case, given the coefficients and the structure of the answer choices, the quadratic formula is the most efficient approach. So, let's dive into it!

The Mighty Quadratic Formula

Alright, let's talk about the star of the show: the quadratic formula. This formula is a magical tool that provides the solutions (also called roots) for any quadratic equation in the standard form. It's derived from the method of completing the square, but the beauty of the formula is that you don't have to go through the steps of completing the square every time – you can just plug in the coefficients and get the answer! The quadratic formula is expressed as: $x = \frac{-b ${}$ \sqrt{b^2 - 4ac}}{2a}$ See those 'a', 'b', and 'c' in the formula? Those are the same coefficients we identified earlier in the standard form of the quadratic equation! The plus-minus symbol ( ${}$ ) indicates that there are usually two solutions to a quadratic equation – one obtained by adding the square root term and the other by subtracting it. These solutions represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis.

Now, let's break down the formula into its key components. First, we have '-b' in the numerator. This means we take the negative of the coefficient 'b'. Then, we have the square root part, which is often called the discriminant. The discriminant, $b^2 - 4ac$, is a critical part of the formula because it tells us about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If it's zero, there is exactly one real solution (a repeated root). And if it's negative, there are two complex solutions. Finally, we divide the entire numerator by '2a', where 'a' is the coefficient of the $x^2$ term. Remember, it's crucial to follow the order of operations (PEMDAS/BODMAS) when plugging in the values and simplifying the formula to avoid errors. The quadratic formula is not just a formula; it's a pathway to understanding the solutions of quadratic equations, providing a systematic way to find the roots regardless of the equation's complexity. So, let's see how we can apply this powerful formula to our specific problem!

Applying the Formula to Our Equation

Okay, guys, the moment we've been waiting for! Let's take the quadratic formula and apply it to our equation, $4x^2 + 3x - 5 = 0$. Remember, we've already identified our coefficients: a = 4, b = 3, and c = -5. Now, it's just a matter of carefully plugging these values into the quadratic formula: $x = \frac-b ${}$ \sqrt{b^2 - 4ac}}{2a}$ Substitute a = 4, b = 3, and c = -5 $x = \frac{-3 ${$ \sqrt{3^2 - 4(4)(-5)}}{2(4)}$ See? It's like filling in the blanks! We've replaced the letters with the corresponding numbers from our equation. This is a crucial step, so double-check your substitutions to make sure everything is in the right place. A small mistake here can lead to a completely wrong answer.

Now, let's simplify the expression step by step. First, focus on the discriminant (the part under the square root): $3^2 - 4(4)(-5) = 9 + 80 = 89$ So, the discriminant is 89. This tells us that we'll have two distinct real solutions because 89 is positive. Next, let's simplify the denominator: $2(4) = 8$ Now, we can rewrite the entire expression as: $x = \frac{-3 ${}$ \sqrt{89}}{8}$ This is the simplified form of the expression that gives us the solutions to our quadratic equation. Notice that we haven't actually calculated the square root of 89 or found the final decimal solutions yet. The question asks us which expression could be used to solve the equation, so we're looking for the expression that matches this form. In many cases, you might need to simplify further to get the exact solutions, but for this question, we're just aiming to identify the correct setup using the quadratic formula. By carefully substituting the coefficients and simplifying the expression, we're one step closer to finding the correct answer choice!

Analyzing the Answer Choices

Alright, let's put on our detective hats and carefully analyze the answer choices provided in the question. We're looking for the expression that matches the one we derived using the quadratic formula: $x = \frac{-3 ${}$ \sqrt{3^2 - 4(4)(-5)}}{2(4)}$ Remember, we plugged in a = 4, b = 3, and c = -5 into the formula. Let's go through the options one by one and see which one fits the bill:

A. $\frac{4 \pm \sqrt{4^2 - 4(3)(-5)}}{2(4)}$ In this option, the -b term is incorrect. It should be -3, but it's given as 4. Also, the value under the square root seems to have the coefficients mixed up. So, this option is not the correct one.

B. $\frac{-4 \pm \sqrt{4^2 - 4(3)(-5)}}{2(4)}$ This option also has an incorrect -b term. It shows -4 instead of -3. The discriminant also appears to have the coefficients mixed up. Therefore, this option is not the correct one either.

C. $\frac{3 \pm \sqrt{3^2 - 4(4)(-5)}}{2(4)}$ Here, the -b term is missing the negative sign. It should be -3, but it's given as 3. The rest of the expression, however, looks promising. The discriminant and the denominator seem to be correctly set up. But, because of the missing negative sign, this option is incorrect.

D. $\frac{-3 \pm \sqrt{3^2 - 4(4)(-5)}}{2(4)}$ Bingo! This option looks like a perfect match! The -b term is -3, the discriminant $3^2 - 4(4)(-5)$ is correctly set up with a = 4, b = 3, and c = -5, and the denominator is 2(4). This is exactly the expression we derived when we applied the quadratic formula to our equation.

By carefully comparing each answer choice to the expression we obtained using the quadratic formula, we can confidently identify the correct answer. It's like a puzzle, where each piece (the coefficients) needs to fit in the right place. This step-by-step analysis is key to avoiding careless mistakes and ensuring that you choose the correct solution.

The Correct Answer and Why

Drumroll, please! After our detailed analysis, it's clear that the correct answer is D. $\frac{-3 \pm \sqrt{3^2 - 4(4)(-5)}}{2(4)}$. Why is this the correct answer? Let's recap:

The Quadratic Formula: We started with the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which is the fundamental tool for solving quadratic equations in the form $ax^2 + bx + c = 0$.
Identifying Coefficients: We correctly identified the coefficients in our equation, $4x^2 + 3x - 5 = 0$, as a = 4, b = 3, and c = -5. This is a critical step because these values are the building blocks for our solution.
Substituting Values: We carefully substituted these values into the quadratic formula, making sure to pay attention to signs and order of operations. This gave us the expression: $x = \frac{-3 \pm \sqrt{3^2 - 4(4)(-5)}}{2(4)}$.
Matching the Expression: We then compared this expression to the answer choices, looking for an exact match. Option D was the only one that perfectly aligned with our derived expression.

The other answer choices were incorrect because they had errors in either the -b term or the discriminant, or both. These errors highlight the importance of carefully substituting the values and paying close attention to the signs. A small mistake can lead to a completely different expression and an incorrect solution.

By understanding the quadratic formula and methodically applying it to the equation, we were able to confidently identify the correct expression. This process not only gives us the answer to this specific question but also reinforces our understanding of how to solve quadratic equations in general. So, pat yourselves on the back, guys! You've successfully navigated the world of quadratic equations!

Final Thoughts and Tips

Great job, everyone! We've successfully tackled the question of which expression could be used to solve $4x^2 + 3x - 5 = 0$. We not only found the correct answer (option D) but also reinforced our understanding of quadratic equations and the powerful quadratic formula. Before we wrap up, let's go over some final thoughts and tips to help you master these types of problems:

Memorize the Quadratic Formula: This is your best friend when it comes to solving quadratic equations. Make sure you know it inside and out: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Write it down several times, practice using it, and soon it will become second nature.
Identify Coefficients Carefully: Pay close attention to the signs of the coefficients a, b, and c. A simple sign error can throw off your entire solution. Double-check your values before plugging them into the formula.
Simplify Step-by-Step: Break down the problem into smaller, manageable steps. Simplify the discriminant first, then the denominator, and finally the entire expression. This will help you avoid errors and keep your work organized.
Check Your Answer: If you have time, plug the solutions you find back into the original equation to make sure they work. This is a great way to catch any mistakes you might have made along the way.
Practice, Practice, Practice: The more you practice, the more comfortable you'll become with quadratic equations. Work through different types of problems, and don't be afraid to make mistakes – they're learning opportunities!

Remember, quadratic equations might seem challenging at first, but with a solid understanding of the concepts and consistent practice, you can conquer them. Keep up the great work, and you'll be solving quadratic equations like a pro in no time! If you have any more questions or want to explore other math topics, feel free to ask. Keep learning and keep growing, guys!