Solving Quadratic Equations: Step-by-Step Guide

Hey math enthusiasts! Today, we're going to dive into the world of quadratic equations and figure out how to solve them. Specifically, we'll tackle the equation $x^2 - x - 14 = 6$. Don't worry if this looks a bit intimidating at first; we'll break it down into easy-to-follow steps. Quadratic equations pop up everywhere in math and science, from calculating the trajectory of a ball to designing the perfect curve. So, understanding how to solve them is a super valuable skill. Let's get started and unravel this math mystery together! This guide will provide a clear, concise, and detailed explanation of how to solve this particular equation, ensuring you grasp the underlying principles and techniques involved. We'll cover everything from the initial setup to the final solutions, making sure you feel confident in your ability to solve similar problems in the future. Ready to become a quadratic equation solver? Let's go!

Step 1: Rearrange the Equation

Alright guys, the first thing we need to do is get our equation into a standard form. Standard form for a quadratic equation looks like this: $ax^2 + bx + c = 0$. This means we need to move everything to one side of the equation, leaving zero on the other side. Our original equation is $x^2 - x - 14 = 6$. To get it into standard form, we need to subtract 6 from both sides. This gives us:

$x^2 - x - 14 - 6 = 6 - 6$

Which simplifies to:

$x^2 - x - 20 = 0$

Now, our equation is in the standard form, and we're ready to move on to the next step. This rearrangement is crucial because it allows us to identify the coefficients $a$, $b$, and $c$, which are essential for solving the equation using various methods. Make sure you don't skip this step, as it sets the foundation for the rest of the solution process. Remember, the goal here is to get all the terms on one side so we can find the values of x that make the equation true. Getting the equation in the right format is like setting up a puzzle; without the correct pieces in place, it's impossible to solve!

Step 2: Choose Your Solution Method

Now that we've got our equation in standard form, we have a few different ways we can solve it. The most common methods include factoring, using the quadratic formula, or completing the square. Each method has its own advantages and disadvantages. For this particular equation, factoring is often the easiest approach, but let's quickly discuss all the options:

• Factoring: This involves breaking down the quadratic expression into two binomial factors. If we can find two numbers that multiply to give us c (in our case, -20) and add up to give us b (which is -1), then we can factor the equation. This method is quick and straightforward if the factors are easily identifiable.

• Quadratic Formula: The quadratic formula is a universal solution for any quadratic equation. It's given by:

  x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  This formula always works, regardless of whether the equation can be factored or not. It's a bit more involved, but it's a surefire way to find the solutions.

• Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. It's a bit more complex, but it's useful for understanding the structure of quadratic equations and is the basis for the quadratic formula.

Since factoring looks like the most straightforward route for $x^2 - x - 20 = 0$, that's what we'll try first. If factoring doesn't work, we can always use the quadratic formula. Knowing multiple methods gives you flexibility and a deeper understanding of the problem. Choosing the right method can save you time and effort. So, let's roll with factoring and see if we can solve this quickly.

Step 3: Factoring the Quadratic Equation

Alright, let's get our hands dirty and try to factor the quadratic expression $x^2 - x - 20$. We're looking for two numbers that multiply to -20 and add up to -1. After a bit of thought, we find that the numbers -5 and 4 fit the bill. Because -5 multiplied by 4 equals -20, and -5 plus 4 equals -1.

So, we can factor the equation as follows:

$(x - 5)(x + 4) = 0$

This means that either $(x - 5) = 0$ or $(x + 4) = 0$. When a product of two factors equals zero, one or both of the factors must be zero. This is the zero-product property, and it's super important in solving factored quadratic equations. This property allows us to break down the problem into two simpler equations, making it much easier to solve for x. Now, let's solve for x in each of these equations.

Step 4: Solve for x

Now we have our factored equation: $(x - 5)(x + 4) = 0$. We'll set each factor equal to zero and solve for x:

1. Solve for x in (x - 5) = 0:

   Add 5 to both sides:

   $x - 5 + 5 = 0 + 5$

   Which gives us:

   $x = 5$

2. Solve for x in (x + 4) = 0:

   Subtract 4 from both sides:

   $x + 4 - 4 = 0 - 4$

   Which gives us:

   $x = -4$

So, we've found our two solutions for x: x = 5 and x = -4. These are the values that, when plugged back into the original equation $x^2 - x - 14 = 6$, will make the equation true. We have successfully solved the quadratic equation by factoring. High five! Now, let's write out the final answer and wrap things up.

Step 5: State the Solutions

We've worked through all the steps and arrived at our solutions. The solutions to the equation $x^2 - x - 14 = 6$ are:

$x = 5$ and $x = -4$

You can always check your answers by plugging these values back into the original equation and ensuring that the equation holds true. This is a crucial step to ensure the accuracy of your solutions and catch any potential errors along the way. Congrats! You've successfully solved a quadratic equation. Keep practicing, and you'll become a pro in no time. Solving quadratic equations is a fundamental skill in algebra and is a stepping stone to more advanced math concepts. Remember, the more you practice, the better you'll get. Keep up the great work, and don't hesitate to revisit these steps anytime you need a refresher. You've got this!

Conclusion: Mastering Quadratic Equations

Wow, guys! We've made it through solving the quadratic equation $x^2 - x - 14 = 6$ step-by-step. We started by rearranging the equation into standard form, then we chose our method (factoring in this case), factored the quadratic expression, solved for x, and finally, stated our solutions: $x = 5$ and $x = -4$. Remember that understanding each step is key. The ability to manipulate and solve quadratic equations is a fundamental skill in mathematics. It is used extensively in a wide variety of scientific and engineering applications. Don't be afraid to try different methods and to practice as much as you can. Math can be tricky, but with perseverance and practice, you can master these concepts. Every step we took today builds a solid foundation for more complex mathematical ideas. Continue to practice and explore, and you'll see your skills grow. Keep asking questions, keep learning, and keep the math excitement alive! You are now well-equipped to tackle similar problems. Good job!