Solving Quadratic Equations Find The Solution Set Of (x+8)(x+8)=0
Hey guys! Today, we're diving into the exciting world of quadratic equations, and we're going to tackle a specific problem that looks pretty straightforward but packs some essential concepts. Our mission, should we choose to accept it, is to find the solution set for the equation (x+8)(x+8)=0. Don't worry, it's not as daunting as it might seem. We'll break it down step by step, making sure everyone, from math newbies to seasoned pros, can follow along. So, grab your pencils, notebooks, or maybe just your favorite note-taking app, and let's get started!
Understanding Quadratic Equations
So, what exactly are we dealing with here? This equation, (x+8)(x+8)=0, is a quadratic equation in disguise. To truly appreciate it, let's first break down the basics of quadratic equations.
What is a Quadratic Equation?
At its core, a quadratic equation is a polynomial equation of the second degree. That might sound like a mouthful, but it simply means the highest power of the variable (in our case, x) is 2. The general form of a quadratic equation is often written as: ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero (because if a were zero, it would no longer be a quadratic equation but a linear one).
In this general form:
- ax² is the quadratic term.
- bx is the linear term.
- c is the constant term.
Now, why do we care about quadratic equations? Well, they pop up in all sorts of real-world situations, from calculating the trajectory of a ball thrown in the air to designing bridges and even in financial modeling. Knowing how to solve them is a seriously useful skill!
Why Solve Quadratic Equations?
The solutions to a quadratic equation, also known as its roots or zeros, are the values of x that make the equation true. In graphical terms, these solutions represent the points where the parabola (the graph of a quadratic equation) intersects the x-axis. Finding these points can help us understand the behavior of various systems and make predictions.
Methods for Solving Quadratic Equations
There are several methods we can use to solve quadratic equations, each with its own strengths and when it's most applicable:
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Factoring: This method involves breaking down the quadratic expression into a product of two binomials. It's super efficient when the equation can be factored easily.
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Completing the Square: This technique transforms the equation into a perfect square trinomial, making it easier to solve. It's a bit more involved but works for all quadratic equations.
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Quadratic Formula: This is the ultimate tool in our arsenal. The quadratic formula can solve any quadratic equation, no matter how messy it looks. It's derived from the method of completing the square and is given by:
x = [ -b ± √(b² - 4ac) ] / 2a
Where a, b, and c are the coefficients from the general form ax² + bx + c = 0.
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Graphical Methods: Graphing the quadratic equation and finding the x-intercepts can also give us the solutions. While not always precise, it provides a visual understanding of the roots.
For our specific problem, (x+8)(x+8)=0, we'll be focusing on the factoring method because it's the most direct and efficient way to solve it.
Expanding the Equation
Before we can dive into solving, let's take a moment to understand why (x+8)(x+8)=0 is indeed a quadratic equation. If we expand this expression, we get:
(x+8)(x+8) = x² + 8x + 8x + 64 = x² + 16x + 64
Now, our equation looks like this:
x² + 16x + 64 = 0
Ah-ha! We can see it fits the general form ax² + bx + c = 0, where a = 1, b = 16, and c = 64. So, we're definitely dealing with a quadratic equation. Understanding this transformation is crucial because it helps us connect the factored form to the standard form, making it easier to identify the coefficients if we were to use the quadratic formula. But, we're sticking with factoring for this one because it's simpler and faster.
Solving (x+8)(x+8)=0 by Factoring
Okay, guys, let's get down to the nitty-gritty. Our equation is (x+8)(x+8)=0. The beauty of this equation is that it's already factored for us! This makes our job significantly easier. Remember, the goal is to find the values of x that make this equation true. This factored form gives us a direct route to the solution.
The Zero Product Property
The key to solving this equation lies in a fundamental principle called the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In simpler terms, if A * B = 0, then either A = 0 or B = 0 (or both).
This property is like a magic trick for solving factored equations. It transforms a multiplication problem into a series of simpler equations that we can solve individually. For our equation, (x+8)(x+8)=0, we have two factors: (x+8) and (x+8). According to the Zero Product Property, if their product is zero, then either the first (x+8) must be zero, or the second (x+8) must be zero.
Applying the Zero Product Property
So, let's apply this property to our equation. We set each factor equal to zero:
- x + 8 = 0
- x + 8 = 0
Notice that we have the same factor repeated, which means we'll get the same solution from both equations. This is a clue that we might have a special type of solution called a repeated root.
Now, let's solve these simple linear equations. To isolate x, we subtract 8 from both sides of each equation:
- x + 8 - 8 = 0 - 8 x = -8
- x + 8 - 8 = 0 - 8 x = -8
Voila! We've found our solution. In both cases, we get x = -8. This means that the value of x that makes the equation (x+8)(x+8)=0 true is -8.
Repeated Roots
Since we got the same solution twice, x = -8, we call this a repeated root or a double root. What does this mean graphically? It means that the parabola representing the equation touches the x-axis at only one point, specifically at x = -8. It doesn't cross the x-axis; it just kisses it and turns around.
Repeated roots are important because they tell us about the nature of the solutions. In this case, it indicates that the parabola has its vertex (the turning point) on the x-axis. This is a key characteristic that helps us understand and visualize the quadratic equation's behavior.
The Solution Set
Now, let's express our solution in the proper notation. The solution set is the set of all values of x that satisfy the equation. In our case, the only value that satisfies (x+8)(x+8)=0 is x = -8. So, we write the solution set as:
{-8}
That's it! We've successfully found the solution set for the equation (x+8)(x+8)=0. We used the Zero Product Property to break down the factored equation into simpler parts and solved for x. Remember, the solution set is a concise way of representing all the solutions to the equation. In this case, we have a single solution, a repeated root, which tells us something special about the quadratic equation's graph.
Alternative Methods and Insights
Okay, we've nailed the factoring method for solving (x+8)(x+8)=0, but let's explore some alternative approaches and gain deeper insights into this equation. Remember, the more tools we have in our math toolbox, the better equipped we are to tackle any problem!
1. Expanding and Using the Quadratic Formula
As we discussed earlier, we can expand the equation (x+8)(x+8)=0 to its standard quadratic form:
x² + 16x + 64 = 0
Now, we can apply the mighty quadratic formula:
x = [ -b ± √(b² - 4ac) ] / 2a
In our case, a = 1, b = 16, and c = 64. Plugging these values into the formula, we get:
x = [ -16 ± √(16² - 4 * 1 * 64) ] / (2 * 1) x = [ -16 ± √(256 - 256) ] / 2 x = [ -16 ± √0 ] / 2 x = -16 / 2 x = -8
See? We arrive at the same solution, x = -8, but through a different route. The quadratic formula is a reliable workhorse, especially when factoring isn't straightforward. Notice how the term under the square root, (b² - 4ac), which is called the discriminant, is zero in this case. A zero discriminant indicates that the quadratic equation has exactly one real solution (a repeated root), which confirms what we found earlier.
2. Completing the Square
Another method we can use is completing the square. While it might seem a bit more involved, it's a powerful technique for solving quadratic equations and is the foundation for deriving the quadratic formula itself. Let's see how it works for our equation:
x² + 16x + 64 = 0
Notice that the left side of the equation is already a perfect square trinomial! It's (x + 8)². So, we can rewrite the equation as:
(x + 8)² = 0
Now, take the square root of both sides:
√(x + 8)² = √0 x + 8 = 0
Finally, subtract 8 from both sides:
x = -8
Again, we get x = -8. Completing the square highlighted that our equation was already in a perfect square form, making the solution even clearer.
3. Graphical Interpretation
Let's visualize our equation graphically. The equation x² + 16x + 64 = 0 represents a parabola. The solutions to the equation are the x-intercepts of the parabola, the points where the parabola intersects the x-axis. In our case, the parabola is given by the function:
y = x² + 16x + 64
If we were to graph this parabola, we would see that it touches the x-axis at only one point: x = -8. This confirms our finding of a repeated root. The vertex of the parabola (the turning point) lies exactly on the x-axis at (-8, 0). This graphical understanding reinforces the concept of a repeated root and gives us a visual way to interpret the solution.
4. Recognizing the Pattern
Our original equation, (x+8)(x+8)=0, can also be written as:
(x + 8)² = 0
This form highlights that we are looking for a number that, when added to 8 and then squared, equals zero. The only number that satisfies this condition is -8, because (-8 + 8)² = 0² = 0. Recognizing this pattern can sometimes lead to a quick mental solution, especially in simpler cases.
Insights and Takeaways
- Multiple Methods: We've seen that there are multiple ways to solve the same quadratic equation. Factoring, the quadratic formula, completing the square, and graphical methods all lead us to the same solution. Choosing the most efficient method depends on the specific equation and your comfort level with each technique.
- The Discriminant: The discriminant (b² - 4ac) in the quadratic formula provides valuable information about the nature of the solutions. A positive discriminant indicates two distinct real solutions, a zero discriminant indicates one real solution (a repeated root), and a negative discriminant indicates no real solutions (two complex solutions).
- Repeated Roots: A repeated root means the parabola touches the x-axis at only one point. This is a special case that tells us the vertex of the parabola lies on the x-axis.
- Graphical Connection: Visualizing quadratic equations as parabolas helps us understand the meaning of the solutions (x-intercepts) and the impact of repeated roots.
Conclusion
Alright, guys! We've successfully navigated the world of quadratic equations and found the solution set for (x+8)(x+8)=0. We started with a simple factored equation, applied the Zero Product Property, and discovered that x = -8 is the magic number. We also explored alternative methods like expanding and using the quadratic formula, completing the square, and graphical interpretation, each reinforcing our solution and providing deeper insights.
Remember, math isn't just about finding the right answer; it's about understanding the process and the underlying concepts. By exploring different approaches and connecting the algebraic solutions to graphical representations, we build a more robust understanding of quadratic equations.
So, next time you encounter a quadratic equation, don't fret! You have a toolbox full of methods to tackle it. Whether it's factoring, the quadratic formula, completing the square, or graphing, you're well-equipped to find those solutions. And remember, practice makes perfect. The more you solve, the more confident you'll become.
Keep exploring, keep learning, and most importantly, keep having fun with math! Until next time, happy solving!