Solving 3x^2 + 8x = -7 A Step-by-Step Guide

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Hey guys! Let's dive into the world of quadratic equations and learn how to solve them by hand. Specifically, we're going to tackle the equation $3x^2 + 8x = -7$. Don't worry, it's not as scary as it looks! We'll break it down step by step, showing all our work and making sure to reduce our answers to their simplest form. So, grab your pencils and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's understand what a quadratic equation actually is. In simple terms, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'x') is 2. The standard form of a quadratic equation is: $ax^2 + bx + c = 0$ Where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation anymore!). Now, you might be wondering, why is this important? Well, quadratic equations pop up everywhere in math and real-world applications. From calculating the trajectory of a ball to designing bridges, they're incredibly useful. The key to solving these equations lies in finding the values of 'x' that make the equation true. These values are also known as the roots or solutions of the equation. We'll use a combination of algebraic manipulation and factoring techniques to find these solutions. Remember, the goal is to isolate 'x' and determine its possible values. This often involves rearranging the equation, combining like terms, and applying the quadratic formula or other methods. So, let's move on and see how we can transform our given equation into the standard form, which is the crucial first step in solving it. We need to get all the terms on one side and set the equation equal to zero. This will allow us to apply the factoring or quadratic formula techniques more easily. Now, let's roll up our sleeves and get our hands dirty with some actual calculations!

Step 1: Convert to Standard Form

The first thing we need to do is convert our equation, $3x^2 + 8x = -7$, into the standard form: $ax^2 + bx + c = 0$. This involves moving all the terms to one side of the equation. Currently, we have -7 on the right side, which we need to bring over to the left. To do this, we simply add 7 to both sides of the equation. This is based on the fundamental principle of algebra that states we can perform the same operation on both sides of an equation without changing its balance. Adding 7 to both sides gives us: $3x^2 + 8x + 7 = -7 + 7$ Simplifying the right side, we get: $3x^2 + 8x + 7 = 0$ Now, our equation is in the standard form, where a = 3, b = 8, and c = 7. This form is essential because it allows us to easily identify the coefficients and constant term, which are necessary for applying various solution methods, such as factoring or the quadratic formula. Converting to standard form might seem like a small step, but it's a crucial foundation for the rest of the solution process. It organizes the equation in a way that makes it easier to manipulate and solve. Without this step, we might struggle to apply the correct methods or identify the correct solutions. So, always remember to start by converting to standard form when solving quadratic equations! This sets the stage for a smooth and accurate solution.

Step 2: Attempt to Factor

Okay, now that our equation is in the standard form ($3x^2 + 8x + 7 = 0$), let's try to factor it. Factoring is a method where we try to express the quadratic expression as a product of two binomials. If we can successfully factor the expression, we can easily find the solutions by setting each factor equal to zero. To factor the quadratic expression $3x^2 + 8x + 7$, we need to find two binomials of the form (px + q)(rx + s) such that when multiplied, they give us the original quadratic expression. This involves finding two numbers that multiply to give the product of the leading coefficient (3) and the constant term (7), which is 21, and add up to the middle coefficient (8). Let's think about the factors of 21: 1 and 21, 3 and 7. None of these pairs add up to 8. This means that the quadratic expression $3x^2 + 8x + 7$ cannot be factored using simple integer coefficients. When we encounter a quadratic expression that doesn't factor easily, it doesn't mean there are no solutions. It simply means that we need to explore other methods to find the roots of the equation. Factoring is a quick and efficient method when it works, but it's not always applicable. So, what do we do when factoring fails? Well, that's where the quadratic formula comes in handy. It's a powerful tool that can solve any quadratic equation, regardless of whether it can be factored or not. Let's move on to the next step and learn how to use the quadratic formula to find the solutions to our equation.

Step 3: Apply the Quadratic Formula

Since we couldn't factor the equation, we're going to use the quadratic formula. This formula is a lifesaver for solving quadratic equations that are difficult or impossible to factor. The quadratic formula is: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where 'a', 'b', and 'c' are the coefficients from our standard form equation ($ax^2 + bx + c = 0$). In our case, we have a = 3, b = 8, and c = 7. Now, let's plug these values into the quadratic formula $x = \frac{-8 \pm \sqrt{8^2 - 4 * 3 * 7}2 * 3}$ First, we simplify the expression under the square root $8^2 - 4 * 3 * 7 = 64 - 84 = -20$ Now, we have: $x = \frac{-8 \pm \sqrt{-20}6}$ Notice that we have a negative number under the square root. This means that the solutions will be complex numbers. Remember, the square root of a negative number involves the imaginary unit 'i', where $i = \sqrt{-1}$. So, we can rewrite $\\sqrt{-20}$ as $\\sqrt{20} * \\sqrt{-1} = \\sqrt{4 * 5} * i = 2i\\sqrt{5}$. Now, our equation looks like this $x = \frac{-8 \pm 2i\\sqrt{5}6}$ We can simplify this further by dividing both the numerator and the denominator by 2 $x = \frac{-4 \pm i\\sqrt{5}3}$ So, our solutions are $x = \frac{-4 + i\\sqrt{5}{3}$ and $x = \frac{-4 - i\\sqrt{5}}{3}$ These are the complex solutions to our quadratic equation. The quadratic formula is a powerful tool because it works for any quadratic equation, regardless of whether it can be factored or not. It guarantees that we can find the solutions, even if they are complex numbers.

Step 4: State the Solutions

Alright, we've made it to the final step! After applying the quadratic formula and simplifying, we've found the solutions to the equation $3x^2 + 8x = -7$. Remember, our solutions are complex numbers because we had a negative value under the square root in the quadratic formula. Our solutions are: $x = \frac{-4 + i\\sqrt{5}}{3}$ and $x = \frac{-4 - i\\sqrt{5}}{3}$ These are the two values of 'x' that satisfy the original equation. Since these solutions are complex, it means that the parabola represented by the quadratic equation does not intersect the x-axis in the real number plane. In other words, there are no real roots for this equation. When stating the solutions, it's important to present them clearly and explicitly. This ensures that anyone reviewing your work can easily identify the answers. We've shown all our steps, from converting to standard form to applying the quadratic formula and simplifying. This thorough approach not only helps us find the correct solutions but also demonstrates our understanding of the process. Solving quadratic equations by hand can seem challenging at first, but with practice and a step-by-step approach, it becomes much easier. Remember to always convert to standard form, try factoring first, and if that doesn't work, use the quadratic formula. And most importantly, don't be afraid of complex numbers! They are just as valid solutions as real numbers. So, keep practicing, and you'll become a pro at solving quadratic equations in no time!

Conclusion

So there you have it, guys! We've successfully solved the quadratic equation $3x^2 + 8x = -7$ by hand. We walked through each step, from converting the equation to standard form to applying the quadratic formula and simplifying the result. Remember, the key to mastering these problems is practice. The more you work through different quadratic equations, the more comfortable you'll become with the process. Don't be discouraged if you encounter complex solutions – they are a natural part of solving quadratic equations, especially when the discriminant (the part under the square root in the quadratic formula) is negative. Keep practicing your factoring skills, and don't hesitate to use the quadratic formula when factoring doesn't work. With consistent effort, you'll be solving quadratic equations like a pro in no time! And remember, math can be fun, so embrace the challenge and enjoy the journey of learning. You got this!