Solve X² - 6x - 7 = 0: Factoring Explained
Hey guys! Let's dive into the fascinating world of quadratic equations. Today, we're going to break down how to solve the equation x² - 6x - 7 = 0. Don't worry, it's not as scary as it looks! We'll go through each step together, making sure you understand the why behind the how. So, grab your pencils and let's get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's quickly recap what a quadratic equation actually is. In its simplest form, a quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants (numbers) and 'a' is not equal to zero. The 'x' is our variable, the thing we're trying to find the value(s) of that make the equation true. The highest power of 'x' in a quadratic equation is 2, which is what gives it the name “quadratic.”
Now, why are quadratic equations important? Well, they pop up all over the place in math, science, and engineering. They can be used to model things like the trajectory of a ball, the shape of a satellite dish, or even the optimal dimensions for a garden. Understanding how to solve them opens up a whole new world of problem-solving possibilities.
There are several methods we can use to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and weaknesses, and the best one to use often depends on the specific equation you're dealing with. In our case, we're going to use the factoring method, which is a super efficient way to solve certain types of quadratic equations, especially when the coefficients are relatively small integers, like in our example x² - 6x - 7 = 0.
Factoring is like reverse multiplication. We're trying to find two binomials (expressions with two terms) that, when multiplied together, give us our original quadratic equation. This method relies on our ability to recognize patterns and break down the equation into its constituent parts. It's a bit like solving a puzzle, and it can be really satisfying when you get it right.
Step-by-Step Solution: Factoring x² - 6x - 7 = 0
Okay, let's get down to business and solve our equation: x² - 6x - 7 = 0. We're going to use the factoring method, so here's how it works:
1. Identify the Coefficients
First, we need to identify the coefficients 'a', 'b', and 'c' in our equation. Remember the standard form: ax² + bx + c = 0. In our case:
- a = 1 (the coefficient of x²)
- b = -6 (the coefficient of x)
- c = -7 (the constant term)
It's crucial to get these right, as they're the building blocks for the next steps. Pay close attention to the signs (positive or negative) because they matter a lot in factoring.
2. Find Two Numbers That Multiply to 'c' and Add Up to 'b'
This is the heart of the factoring method. We need to find two numbers that, when multiplied together, give us 'c' (-7 in our case), and when added together, give us 'b' (-6). This might sound tricky, but with a little practice, you'll get the hang of it.
Let's think about the factors of -7. Since 7 is a prime number, its only factors are 1 and 7. To get a negative product (-7), one of the factors must be negative. So, our possible pairs of factors are (1, -7) and (-1, 7). Now, let's see which pair adds up to -6:
- 1 + (-7) = -6 (Bingo!)
- -1 + 7 = 6 (Nope)
So, the two numbers we're looking for are 1 and -7.
3. Rewrite the Quadratic Equation
Now that we've found our two numbers, we can rewrite the middle term (-6x) in our equation using these numbers. This is the key step in factoring.
Our original equation is x² - 6x - 7 = 0. We're going to rewrite the -6x term as 1x - 7x. So, our equation becomes:
x² + 1x - 7x - 7 = 0
Notice that we haven't changed the value of the equation; we've just rewritten it in a way that will allow us to factor it.
4. Factor by Grouping
Now comes the fun part: factoring by grouping. We're going to group the first two terms and the last two terms together and factor out the greatest common factor (GCF) from each group.
Our equation is x² + 1x - 7x - 7 = 0. Let's group the terms:
(x² + 1x) + (-7x - 7) = 0
Now, let's factor out the GCF from each group:
- From (x² + 1x), the GCF is x. Factoring out x gives us x(x + 1).
- From (-7x - 7), the GCF is -7. Factoring out -7 gives us -7(x + 1).
So, our equation now looks like this:
x(x + 1) - 7(x + 1) = 0
Notice something cool? We have a common factor of (x + 1) in both terms! This is a good sign that we're on the right track.
5. Factor Out the Common Binomial
Since (x + 1) is a common factor, we can factor it out from the entire equation. This is the final step in the factoring process.
Our equation is x(x + 1) - 7(x + 1) = 0. Factoring out (x + 1) gives us:
(x + 1)(x - 7) = 0
Ta-da! We've factored our quadratic equation. We've successfully rewritten it as the product of two binomials.
6. Set Each Factor to Zero and Solve for x
Now that we've factored the equation, we can 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 A * B = 0, then either A = 0 or B = 0 (or both).
In our case, we have (x + 1)(x - 7) = 0. So, we can set each factor equal to zero and solve for x:
- x + 1 = 0
- Subtract 1 from both sides: x = -1
- x - 7 = 0
- Add 7 to both sides: x = 7
7. The Solutions!
We've done it! We've found the solutions to our quadratic equation x² - 6x - 7 = 0. The solutions are:
- x = -1
- x = 7
These are the values of 'x' that make the equation true. If you plug either of these values back into the original equation, you'll see that it equals zero. Awesome, right?
Checking Our Answers
It's always a good idea to check your answers, especially in math. This helps you catch any mistakes and build confidence in your solutions. Let's check our solutions x = -1 and x = 7 by plugging them back into the original equation x² - 6x - 7 = 0.
Checking x = -1
Substitute x = -1 into the equation:
(-1)² - 6(-1) - 7 = 0
Simplify:
1 + 6 - 7 = 0
0 = 0
Our solution x = -1 checks out! The equation holds true.
Checking x = 7
Substitute x = 7 into the equation:
(7)² - 6(7) - 7 = 0
Simplify:
49 - 42 - 7 = 0
0 = 0
Our solution x = 7 also checks out! We've got it right.
Key Takeaways and Practice Makes Perfect
So, there you have it! We've successfully solved the quadratic equation x² - 6x - 7 = 0 using the factoring method. Let's recap the key steps:
- Identify the coefficients: Find 'a', 'b', and 'c'.
- Find two numbers: That multiply to 'c' and add up to 'b'.
- Rewrite the equation: Using the two numbers to split the middle term.
- Factor by grouping: Factor out the GCF from each pair of terms.
- Factor out the common binomial: The binomial should be the same in both terms.
- Set each factor to zero: Use the zero product property.
- Solve for x: Find the solutions to the equation.
Remember, practice makes perfect! The more quadratic equations you solve, the more comfortable you'll become with the factoring method and other techniques. Don't be afraid to make mistakes; they're a natural part of the learning process. Keep practicing, and you'll be solving quadratic equations like a pro in no time!
Quadratic equations are a fundamental concept in algebra, and mastering them will give you a strong foundation for more advanced math topics. Understanding the factoring method is crucial because it’s a quick and efficient way to solve many quadratic equations, especially those with integer coefficients. However, it's not always the best method for every equation. Sometimes, you might encounter equations that are difficult or impossible to factor using integers. That's where other methods, like completing the square or the quadratic formula, come in handy. We might explore those methods in future discussions, so stay tuned!
In conclusion, solving quadratic equations is a valuable skill that opens doors to various mathematical and real-world applications. By understanding the factoring method and practicing regularly, you'll be well-equipped to tackle a wide range of problems. So, keep up the great work, and remember, math can be fun!
