Finding The Missing Constant: Solving Consecutive Integer Problems

Diving into the Problem: Understanding Consecutive Integers

Hey guys! Let's break down this math problem step by step. The core of the question revolves around consecutive integers and the fascinating world of quadratic equations. The problem starts with a simple statement: "The product of two consecutive integers is 420." That's our starting point. The goal? To figure out the constant term in a quadratic equation that helps us solve this. So, what exactly are consecutive integers? They're simply whole numbers that follow each other in order, like 1 and 2, 10 and 11, or even 100 and 101. The key here is that they differ by exactly one. When we're dealing with consecutive integers in an algebraic problem, a clever trick we use is to represent them with variables. We can call the smaller integer 'x'. Since the next integer is always one more than the smaller one, we can represent the larger integer as 'x + 1'. Now, remember that the product of these two consecutive integers is 420. This translates to the following equation: x * (x + 1) = 420. This is where things start to get interesting because we are looking for the unknown that will fit into the standard form of the quadratic equation: x² + x + ☐ = 0. From here, we will learn how to manipulate the terms to identify and finally calculate the constant. It's like a mathematical puzzle! The first step in solving this problem involves setting up the equation. This requires a clear understanding of what the problem is asking. So, we need to translate the words into a mathematical expression that we can work with. From here, we can expand and work towards completing the quadratic equation to put it into standard form. This will help us to find the missing term.

Let's consider how to solve the initial equation and transform it in order to find the missing constant.

Transforming the Equation: From Product to Quadratic Form

Alright, let's get our hands dirty with some algebra! Our equation, which represents the product of two consecutive integers, is x * (x + 1) = 420. Our goal now is to reshape this equation into the standard form of a quadratic equation, which is ax² + bx + c = 0. In our case, we want the equation to look like x² + x + ☐ = 0, where we need to find the missing constant represented by the box. Here's how we're going to do it, step by step. First, we need to expand the left side of our original equation, which means multiplying 'x' by both 'x' and '1'. That gives us x² + x = 420. See how we're already starting to get closer to our target quadratic form? Now, to get the equation fully into standard form, we need to move the constant term (420) to the left side of the equation. We do this by subtracting 420 from both sides. This gives us x² + x - 420 = 0. And there we have it! We now have a quadratic equation in standard form. Compare this to the form x² + x + ☐ = 0. It's really easy to see where the missing term, which is the constant we are looking for, actually is. Now, can you tell which one it is? You're right! The constant is -420. Therefore, in the equation x² + x + ☐ = 0, the constant is -420. Remember that in a quadratic equation, the constant term is the value that doesn't have a variable attached to it. In the case of x² + x - 420 = 0, the constant term is the -420, which we were able to get in the standard form using algebraic manipulations.

Quick tip:

Always remember that you can check your work by plugging the solution(s) back into the original equation.

Unveiling the Constant: The Final Answer

So, we've journeyed through the problem, starting with consecutive integers, building an equation, and transforming it into standard quadratic form. Now comes the moment of truth: what is the constant of the quadratic expression in this equation? Let's recap what we've done. We began with the idea that the product of two consecutive integers is 420. We then represented the consecutive integers with variables, specifically x and x + 1. The equation we formed was x * (x + 1) = 420. Then we expanded this, which gave us x² + x = 420. Then we put it in the standard form of a quadratic equation, by subtracting 420 from both sides, resulting in x² + x - 420 = 0. Comparing this with the format we are asked to fill in x² + x + ☐ = 0, we can now simply identify the constant. The constant is the number that stands alone in the equation and the number that fills in the blank. In our transformed equation, x² + x - 420 = 0, the constant term is -420. Therefore, when filling in the blank in the equation x² + x + ☐ = 0, the correct value to enter is -420. Congratulations! You have successfully solved the problem and found the missing constant term. This step-by-step breakdown hopefully gave you a clear understanding of how to approach the problem. Now you know the process of how to solve for the missing constant. It’s a great illustration of how mathematical concepts connect and how algebraic manipulation is used to solve complex problems. It's all about the journey of transforming an equation to reach the final solution and identify the missing constant. The beauty of mathematics lies in its ability to break down problems into manageable steps, making even the most complex concepts accessible. Remember, practice makes perfect, so keep exploring, keep learning, and keep asking questions!

Additional Insights: Solving the Equation

Let's go beyond simply finding the constant. We now can even solve for the original consecutive integers. Even though the original question only asked for the constant term, let's quickly solve the equation to get a complete picture. Our standard quadratic equation is x² + x - 420 = 0. One way to solve this is by factoring. We're looking for two numbers that multiply to -420 and add up to 1 (the coefficient of the 'x' term). Those numbers are 21 and -20. So, we can factor the equation as (x + 21)(x - 20) = 0. Setting each factor equal to zero gives us two possible solutions for x: x + 21 = 0, which means x = -21, and x - 20 = 0, which means x = 20. Therefore, the two consecutive integers could be 20 and 21 (since 20 * 21 = 420), or they could be -21 and -20 (since -21 * -20 = 420). This is just a quick demonstration of how to solve the equation from the quadratic form.

Why does it work?

Factoring relies on the distributive property (sometimes called the FOIL method). When we multiply (x + 21)(x - 20), we are essentially undoing the expansion process we did earlier. The factored form allows us to easily identify the values of 'x' that make the equation equal to zero. This is the core principle behind solving quadratic equations by factoring.

What if factoring is difficult?

While factoring is a great way to solve many quadratic equations, sometimes the numbers are not easy to work with. In those cases, the quadratic formula is your best friend! The quadratic formula is a universal tool for solving quadratic equations, guaranteeing a solution regardless of the numbers involved. The quadratic formula is derived by completing the square and can be used for all quadratic equations. Knowing both methods provides flexibility when solving these types of problems. This gives you a wider variety of ways to tackle any quadratic equation you encounter. These methods are essential tools in your math toolbox.