Solving Quadratic Equations Step By Step
Hey guys! Today, we're diving into the world of quadratic equations. Don't worry, it's not as scary as it sounds! We're going to break down how to solve two specific equations, step by step. So grab your pencils, and let's get started!
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what quadratic equations are. In simple terms, 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:
Where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation anymore!). The solutions to a quadratic equation are also known as its roots or zeros, which are the values of 'x' that make the equation true. We will be focusing on solving quadratic equations in this article, and mastering this skill is crucial for various fields like physics, engineering, and even economics. Understanding quadratic equations will help you model projectile motion, optimize areas, and make informed decisions based on mathematical models. So let’s make sure we nail this down, okay? When you’re tackling these problems, always remember that our goal is to rearrange the equation into a standard form. This will usually involve moving terms around to one side, so we have zero on the other side. Then, we can apply our strategies to solve for ‘x’. Speaking of strategies, there are a few key methods we can use. Factoring is one of them, which involves breaking down the quadratic expression into simpler expressions. Another powerful tool is the quadratic formula, which provides a direct way to find solutions, regardless of the complexity of the equation. We’ll be using these techniques as we work through our examples, so keep them in mind! Learning how to approach quadratic equations is like adding a new tool to your problem-solving toolbox. It gives you the ability to handle a wide range of mathematical challenges and to understand the underlying principles behind many real-world phenomena. So, stay engaged, keep practicing, and soon you’ll be solving these like a pro.
A. Solving $x^2 = 4x + 21$
Okay, let's tackle our first equation: $x^2 = 4x + 21$. The first thing we need to do is rearrange the equation into the standard form, which, as we mentioned, is $ax^2 + bx + c = 0$. To do this, we'll subtract $4x$ and 21 from both sides of the equation. This will move all the terms to one side, leaving us with zero on the other side. This is a crucial step because many solution methods, like factoring and using the quadratic formula, require the equation to be in this standard form. By getting the equation into this form, we set ourselves up for success in the next steps. So, let’s see how this looks:
Now that we have the equation in the standard form, we can proceed to solve it. There are a couple of ways we can do this: factoring or using the quadratic formula. For this particular equation, factoring seems like a good approach because we can look for two numbers that multiply to -21 and add up to -4. This method is often quicker if the quadratic expression can be easily factored. Factoring is a great skill to develop because it allows you to break down complex expressions into simpler ones. It's like finding the puzzle pieces that fit together to form the whole picture. When you factor a quadratic expression, you’re essentially reversing the process of expanding brackets. If you can identify the factors quickly, you can save time and avoid more complicated methods like the quadratic formula. So, let’s think about the factors of -21. We have 1 and -21, -1 and 21, 3 and -7, and -3 and 7. Which pair adds up to -4? It’s 3 and -7! This means we can factor the quadratic expression as follows:
Now we've got the equation in a factored form, which is fantastic! The next step is to use the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if we have two expressions multiplied together equaling zero, then we can set each expression equal to zero and solve for 'x'. This is a powerful technique because it transforms one quadratic equation into two simpler linear equations. Linear equations are much easier to solve, typically involving just a few algebraic steps. So, let’s apply the zero-product property to our factored equation. We have (x + 3)(x - 7) = 0, which means either (x + 3) = 0 or (x - 7) = 0. Now we have two simple equations to solve.
So, let’s take each factor and set it to zero:
x + 3 = 0$ or $x - 7 = 0
Now we have two simple equations to solve. For the first equation, $x + 3 = 0$, we subtract 3 from both sides to isolate 'x'. This gives us $x = -3$. For the second equation, $x - 7 = 0$, we add 7 to both sides to isolate 'x', which gives us $x = 7$. And just like that, we have our two solutions! These values of 'x' are the roots of the original quadratic equation. They are the points where the parabola, represented by the quadratic equation, intersects the x-axis. Knowing the roots of a quadratic equation is crucial in many applications, such as finding the zeros of a function or determining the points of equilibrium in a system. We found our solutions by carefully applying algebraic principles and breaking down the problem into manageable steps. This process of simplification and methodical problem-solving is at the heart of mathematics. So, let’s write down our solutions:
Therefore, the solutions for the equation $x^2 = 4x + 21$ are:
B. Solving $3x^2 + 27 = 18x$
Alright, let's move on to our second equation: $3x^2 + 27 = 18x$. Just like before, our first step is to get this equation into the standard form, which is $ax^2 + bx + c = 0$. To do this, we need to move all the terms to one side of the equation, leaving zero on the other side. In this case, we need to subtract $18x$ from both sides of the equation. This will rearrange the terms so that they are in the correct order, with the $x^2$ term first, followed by the $x$ term, and then the constant term. Getting the equation into this standard form is crucial because it allows us to easily identify the coefficients a, b, and c, which we’ll need if we decide to use the quadratic formula. It also sets us up nicely for factoring if that turns out to be a viable option. So, let's go ahead and do that:
Now that we have the equation in the standard form, let's take a closer look. Notice that all the coefficients (3, -18, and 27) are divisible by 3. This means we can simplify the equation by dividing both sides by 3. Simplifying the equation is a smart move because it makes the numbers smaller and easier to work with. It reduces the chances of making mistakes in the subsequent steps, whether we're factoring or applying the quadratic formula. Plus, it doesn't change the solutions of the equation, which is the most important thing. Think of it as cleaning up the problem a bit before diving into the nitty-gritty. So, let's go ahead and divide both sides by 3:
Great! The equation looks much simpler now. Now, we need to solve for 'x'. Let’s think about our options. Factoring might be a good approach here because the coefficients are relatively small and easy to manage. We're looking for two numbers that multiply to 9 and add up to -6. If we can find such numbers, then we can factor the quadratic expression into two binomials. Factoring is often the quickest method when it works, so it’s worth exploring. It’s like finding the right key to unlock the solution. If factoring doesn’t work, then we can always fall back on the quadratic formula, which will work for any quadratic equation. But let’s give factoring a try first. We need to find two numbers that meet our criteria. So, let's see... what numbers multiply to 9 and add to -6? The numbers -3 and -3 fit the bill perfectly! Both conditions are met, which means we’re on the right track. This means we can factor the quadratic expression as follows:
Notice anything special about this factored form? We have the same factor repeated twice! This means that the quadratic expression is a perfect square. Recognizing patterns like this can save us time and effort in the long run. A perfect square trinomial factors into the square of a binomial, which is exactly what we have here. This also tells us something about the solutions to the equation: we’re likely to have a repeated root. This is when the quadratic equation has only one distinct solution. So, now that we have our factored form, let's apply the zero-product property, which we used in the previous problem. Remember, this property tells us that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have the same factor repeated twice, but the principle still applies. So, let’s see how we can use this to find our solution.
Now, using the zero-product property, we set each factor equal to zero:
Since both factors are the same, we only need to solve this equation once. Add 3 to both sides:
And there we have it! We found that $x = 3$ is the solution to the equation. Because we had a repeated factor, this is a repeated root, meaning it’s the only solution to this quadratic equation. This is a slightly different scenario from our first equation, where we had two distinct solutions. In this case, the parabola represented by the quadratic equation touches the x-axis at only one point, which is at $x = 3$. Understanding the nature of the solutions – whether they are distinct, repeated, or even complex – is a key part of mastering quadratic equations. It gives us a deeper insight into the behavior of these equations and the graphs they represent. So, let’s state our final answer:
Therefore, the solution for the equation $3x^2 + 27 = 18x$ is:
Conclusion
And that's a wrap, guys! We've successfully solved two quadratic equations using factoring and the zero-product property. Remember, the key is to get the equation into the standard form first, then choose the best method for solving. Keep practicing, and you'll become a quadratic equation pro in no time! We started with an equation that we could solve by moving terms around and then factoring. We saw how important it is to rearrange the equation into standard form, so we can easily identify the coefficients and apply the right solution method. Then, we tackled another equation that required us to simplify it before solving. We divided both sides by a common factor to make the numbers more manageable. This step highlighted the importance of looking for opportunities to simplify equations, as it can save us time and reduce the chances of making mistakes. Each equation presented its own unique challenges and required us to think strategically about how to approach it. These strategies and techniques you learn will apply to a broad range of mathematical problems. So, keep up the great work, and remember, practice makes perfect. The more equations you solve, the more confident and skilled you’ll become. And who knows? You might even start to enjoy the process of untangling these mathematical puzzles! So, until next time, keep those pencils sharp and your minds even sharper!
