Solving The Quadratic Equation 4x^2 - X - 3 = 0 A Step-by-Step Guide

Hey guys! Are you struggling with quadratic equations? Don't worry; we've all been there! Today, we're going to break down how to solve the equation $4x^2 - x - 3 = 0$. This type of equation is a classic example of a quadratic equation, and mastering it is super important for anyone diving into algebra and beyond. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

First, let's make sure we're all on the same page. A quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our case, we have $4x^2 - x - 3 = 0$, so $a = 4$, $b = -1$, and $c = -3$. The solutions to a quadratic equation are also known as the roots or zeros of the equation. Finding these roots is what we're aiming for. There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. We'll primarily focus on factoring and using the quadratic formula in this guide, as they are the most commonly used and versatile methods. These methods aren't just about finding the right answer; they're about understanding the relationships between the coefficients and the solutions, and about building a strong foundation for more advanced mathematical concepts. Understanding these concepts thoroughly will empower you to tackle a wide range of problems with confidence and precision. We will carefully explore these methods, providing detailed explanations and examples to help you grasp every step. So, stick with us, and you'll be solving quadratic equations like a pro in no time!

Method 1 Factoring the Quadratic Equation

Factoring is a fantastic method when it works because it's often the quickest way to solve a quadratic equation. The idea behind factoring is to rewrite the quadratic equation as a product of two binomials. If we can do that, then we can easily find the solutions. Let's apply this to our equation, $4x^2 - x - 3 = 0$.

Step 1 Find Two Numbers

We need to find two numbers that multiply to $a imes c$ (which is $4 imes -3 = -12$) and add up to $b$ (which is $-1$). After a little thought, we can see that the numbers $-4$ and $3$ fit the bill since $-4 imes 3 = -12$ and $-4 + 3 = -1$. This step is crucial; it's like finding the key pieces of a puzzle. Think of different number pairs and their products and sums. With practice, you'll get quicker at identifying the right pair.

Step 2 Rewrite the Middle Term

Now, we'll rewrite the middle term, $-x$, using the two numbers we found. So, we replace $-x$ with $-4x + 3x$. Our equation becomes $4x^2 - 4x + 3x - 3 = 0$. This rewriting might seem a bit strange at first, but it sets us up perfectly for the next step. It’s all about manipulating the equation without changing its value, so we can factor it more easily. By breaking down the middle term, we create groups that share common factors, which is essential for factoring by grouping.

Step 3 Factor by Grouping

Next, we factor by grouping. We group the first two terms and the last two terms: $(4x^2 - 4x) + (3x - 3) = 0$. Now, we factor out the greatest common factor (GCF) from each group. From the first group, $4x$ is the GCF, and from the second group, $3$ is the GCF. Factoring these out, we get $4x(x - 1) + 3(x - 1) = 0$. Notice that we now have a common binomial factor, $(x - 1)$. This is a clear sign that we're on the right track. Factoring by grouping is like assembling building blocks; you find the common pieces and put them together. This method relies on recognizing patterns and common factors within the equation.

Step 4 Factor out the Common Binomial

We can now factor out the common binomial factor, $(x - 1)$, which gives us $(x - 1)(4x + 3) = 0$. We've successfully factored the quadratic equation into the product of two binomials! This is a major milestone. Factoring out the binomial is the final piece of the puzzle, and it transforms the equation into a form that is easy to solve. By identifying and extracting the common binomial, we simplify the equation and reveal its solutions.

Step 5 Solve for x

To find the solutions for $x$, we set each factor equal to zero: $x - 1 = 0$ and $4x + 3 = 0$. Solving these linear equations is straightforward. For $x - 1 = 0$, we add $1$ to both sides to get $x = 1$. For $4x + 3 = 0$, we subtract $3$ from both sides to get $4x = -3$, and then divide by $4$ to get x = -rac{3}{4}. So, our solutions are $x = 1$ and x = -rac{3}{4}. These are the values of $x$ that make the original equation true. Solving for $x$ is the ultimate goal, and by setting each factor to zero, we unlock the solutions. This step is the culmination of the factoring process, and it provides the answer we've been working towards.

Method 2 Using the Quadratic Formula

When factoring isn't straightforward, or you just want a method that always works, the quadratic formula is your best friend. The quadratic formula is a powerful tool that can solve any quadratic equation, regardless of whether it can be easily factored. It's like having a universal key that unlocks the solutions to any quadratic equation. Let’s see how it works for our equation, $4x^2 - x - 3 = 0$.

Step 1 Recall the Quadratic Formula

The quadratic formula is given by:

x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}

Make sure you memorize this formula! It’s a cornerstone of algebra. The quadratic formula is derived from the method of completing the square, and it provides a direct way to find the roots of a quadratic equation. It's a reliable method that works for all quadratic equations, even those that are difficult or impossible to factor. Having this formula in your mathematical toolkit is essential for solving a wide range of problems.

Step 2 Identify a, b, and c

In our equation, $4x^2 - x - 3 = 0$, we have $a = 4$, $b = -1$, and $c = -3$. Identifying these coefficients correctly is crucial for applying the quadratic formula accurately. Be careful with signs! A mistake here can throw off your entire calculation. These coefficients determine the shape and position of the parabola represented by the quadratic equation, and they play a key role in finding the roots.

Step 3 Plug the Values into the Formula

Now, we substitute these values into the quadratic formula:

x = rac{-(-1) ext{±} ext{√}((-1)^2 - 4(4)(-3))}{2(4)}

Plugging in the values is like loading the formula with the specific details of our equation. Make sure to substitute carefully, paying close attention to parentheses and signs. This step is where the formula transforms from a general solution into a specific solution for our problem. It's a critical step that sets the stage for the arithmetic calculations that will lead us to the roots.

Step 4 Simplify

Let's simplify step by step:

x = rac{1 ext{±} ext{√}(1 + 48)}{8}

x = rac{1 ext{±} ext{√}(49)}{8}

x = rac{1 ext{±} 7}{8}

Simplifying the expression involves a series of arithmetic operations. Start by simplifying the expression under the square root, then evaluate the square root. Continue simplifying the numerator and denominator until you reach the simplest possible form. This step requires careful attention to detail and a solid understanding of arithmetic operations. Breaking down the simplification into smaller steps helps to minimize errors and ensures an accurate result.

Step 5 Find the Two Solutions

Now we have two cases to consider:

1. Using the plus sign: x = rac{1 + 7}{8} = rac{8}{8} = 1
2. Using the minus sign: x = rac{1 - 7}{8} = rac{-6}{8} = -rac{3}{4}

So, the solutions are $x = 1$ and x = -rac{3}{4}, which are the same solutions we found by factoring! This confirms that the quadratic formula is a reliable method for solving quadratic equations. Breaking down the ± sign into two separate calculations allows us to find both roots of the equation. Each root represents a point where the parabola intersects the x-axis. Finding these solutions is the ultimate goal, and it provides valuable information about the behavior of the quadratic equation.

Conclusion

Great job, guys! We've successfully solved the quadratic equation $4x^2 - x - 3 = 0$ using both factoring and the quadratic formula. We found that the solutions are $x = 1$ and x = -rac{3}{4}. Whether you prefer factoring or the quadratic formula, the important thing is to understand both methods and know when to apply them. Remember, practice makes perfect, so keep working on these problems, and you'll become a quadratic equation master in no time! Solving quadratic equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts. Keep practicing, and you'll build confidence and proficiency in your problem-solving abilities.