Solving For X In The Quadratic Equation X² + 14x + 49 = 0
Hey guys! Ever stumbled upon a quadratic equation and felt a bit lost? Don't worry, it happens to the best of us. Today, we're going to break down a classic example and find the value of x in the equation x² + 14x + 49 = 0. We'll explore different methods to solve this, making sure you understand each step along the way. So, let's dive in and make math a little less mysterious!
Understanding Quadratic Equations
Before we jump into solving this particular equation, let's quickly recap what quadratic equations are. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in our case, x) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. These equations pop up in various fields, from physics to engineering, so mastering them is super important.
The solutions to a quadratic equation are also known as its roots or zeros. These are the values of x that make the equation true. A quadratic equation can have two real roots, one real root (which is a repeated root), or two complex roots. Our goal is to find these roots for the equation x² + 14x + 49 = 0. Understanding the nature of roots helps us in predicting the type of solutions we should expect. For instance, the discriminant (which we'll touch on later) can tell us whether the roots are real or complex, and whether they are distinct or repeated. This foundational knowledge is crucial for tackling more complex problems and for appreciating the elegance of algebra.
Methods to Solve Quadratic Equations
There are several methods to solve quadratic equations, but we'll focus on two common ones: factoring and using the quadratic formula. Factoring is often the quickest method if the equation can be easily factored. The quadratic formula, on the other hand, is a foolproof method that works for any quadratic equation. Let's explore both methods in detail.
Factoring the Equation
Factoring involves expressing the quadratic equation as a product of two binomials. This method is efficient when the quadratic expression can be easily factored. For our equation, x² + 14x + 49 = 0, we need to find two numbers that multiply to 49 and add up to 14. Think about it for a moment... what numbers fit the bill? You got it – 7 and 7! So, we can rewrite the equation as (x + 7)(x + 7) = 0. This means (x + 7)² = 0. Setting each factor equal to zero, we get x + 7 = 0. Solving for x, we find x = -7. Since both factors are the same, we have a repeated root. Factoring is a powerful technique because it breaks down a complex problem into simpler parts. It relies on recognizing patterns and applying the distributive property in reverse. Mastering factoring not only helps in solving quadratic equations but also enhances algebraic manipulation skills in general. The key to efficient factoring is practice – the more you do it, the quicker you'll become at spotting the factors.
Using the Quadratic Formula
Now, let's talk about the quadratic formula. This formula is your best friend when factoring seems tricky or impossible. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0. In our case, a = 1, b = 14, and c = 49. Let's plug these values into the formula:
x = (-14 ± √(14² - 4 * 1 * 49)) / (2 * 1)
Simplify the expression step by step:
x = (-14 ± √(196 - 196)) / 2
x = (-14 ± √0) / 2
x = -14 / 2
x = -7
As you can see, we arrived at the same answer, x = -7, using the quadratic formula. The quadratic formula is a universal tool that always works, regardless of whether the equation is easily factorable. It's derived from the method of completing the square, which is another technique for solving quadratic equations. The discriminant, b² - 4ac, plays a crucial role in determining the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it's zero, there is one repeated real root; and if it's negative, there are two complex roots. Understanding the discriminant helps predict the type of solutions you'll get, making it an invaluable part of your problem-solving toolkit.
Analyzing the Solutions
We've found that x = -7 is the solution to the equation x² + 14x + 49 = 0. Let's analyze this result. Since we got the same root using both factoring and the quadratic formula, we can be confident in our answer. Moreover, the fact that we have a repeated root indicates that the quadratic expression is a perfect square trinomial. This means it can be written in the form (x + k)² or (x - k)², where k is a constant. In our case, x² + 14x + 49 is indeed (x + 7)². Analyzing solutions is crucial because it allows us to check for errors and understand the behavior of the equation. For example, if we had obtained two different real roots, we would know that the quadratic expression could not be a perfect square trinomial. Similarly, if we had obtained complex roots, we would know that the graph of the quadratic function does not intersect the x-axis. This type of analysis deepens our understanding and improves our problem-solving skills.
Choosing the Correct Alternative
Now, let's look at the alternatives provided:
a) -7 b) 0 c) 7 d) -14
The correct answer is clearly a) -7. We've solved the equation using two different methods and confirmed that x = -7 is the solution. Choosing the correct alternative is the final step in the problem-solving process. It's important to carefully compare your solution with the given options to avoid careless mistakes. In this case, the other alternatives are incorrect because they do not satisfy the original equation. For example, if we substitute x = 0 into the equation, we get 0² + 14*0 + 49 = 49, which is not equal to zero. Similarly, x = 7 gives 7² + 14*7 + 49 = 49 + 98 + 49 = 196, which is also not equal to zero. This reinforces the importance of verifying your solution by substituting it back into the original equation.
Real-World Applications
Quadratic equations aren't just abstract math problems; they have tons of real-world applications. They are used in physics to describe projectile motion, in engineering to design structures, and in economics to model supply and demand. For instance, if you throw a ball, the path it follows can be modeled by a quadratic equation. Similarly, the shape of a suspension bridge can be described using a quadratic function. Understanding quadratic equations allows us to make predictions and solve problems in these various fields. For example, engineers use quadratic equations to calculate the maximum height and range of a projectile, or to determine the optimal shape of an arch for structural stability. Economists use quadratic models to analyze market trends and predict consumer behavior. These applications highlight the practical significance of mastering quadratic equations and their role in solving real-world problems.
Conclusion
So, there you have it! We've successfully found the value of x in the quadratic equation x² + 14x + 49 = 0. We used both factoring and the quadratic formula to arrive at the solution x = -7. Remember, practice makes perfect, so keep solving those equations! Understanding quadratic equations is a fundamental skill in mathematics, and it opens the door to more advanced topics. Whether you're solving problems in algebra, calculus, or even real-world applications, the ability to work with quadratic equations is essential. Keep practicing, and you'll become more confident and proficient in solving these types of problems. Math can be challenging, but it's also incredibly rewarding when you understand the concepts and can apply them effectively. Keep exploring, keep learning, and you'll continue to grow your mathematical skills!
