Solving Quadratic Equations: A Step-by-Step Guide
Let's dive into solving the quadratic equation x² - 2x = 35. Quadratic equations might seem intimidating at first, but with a bit of algebraic maneuvering, they become quite manageable. In this guide, we'll break down the steps to solve this equation, ensuring you understand each part of the process. We'll also explore the underlying concepts, so you're not just memorizing steps but truly grasping how to tackle similar problems. The goal is to provide you with a clear, concise, and comprehensive approach to solving quadratic equations, making you more confident in your mathematical abilities. This understanding will be invaluable as you advance in mathematics and encounter more complex problems that build upon these fundamental concepts.
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's briefly discuss what quadratic equations are and why they are important. A quadratic equation is a polynomial equation of the second degree. The general form is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The key characteristic is the x² term, which makes it a second-degree equation. Quadratic equations pop up in various real-world scenarios, from physics (projectile motion) to engineering (designing structures) and even economics (modeling growth). Understanding how to solve them is a fundamental skill in many quantitative fields.
The importance of quadratic equations lies in their ability to model parabolic relationships. Imagine throwing a ball in the air; the path it follows can be described by a quadratic equation. Similarly, the curve of a suspension bridge or the trajectory of a rocket can be modeled using quadratic equations. These equations allow us to predict outcomes, optimize designs, and understand the behavior of systems that exhibit parabolic motion or relationships. Moreover, quadratic equations form the basis for more advanced mathematical concepts, such as calculus and differential equations, making their mastery essential for further studies in mathematics and related fields.
The solutions to a quadratic equation are also known as its roots or zeros. These are the values of x that satisfy the equation, i.e., make the equation true. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots. The nature of the roots depends on the discriminant, which is the part of the quadratic formula under the square root sign (b² - 4ac). If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; and if it is negative, there are two complex roots. Understanding the discriminant can provide valuable insights into the nature of the solutions without actually solving the equation. This knowledge can save time and effort in problem-solving and deepen your understanding of quadratic equations.
Step 1: Rearrange the Equation
Our initial equation is x² - 2x = 35. To solve it, we first need to set it to the standard quadratic form, which is ax² + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. To do this, we subtract 35 from both sides of the equation:
x² - 2x - 35 = 0
Now, our equation is in the standard quadratic form, where a = 1, b = -2, and c = -35. This form is crucial because it allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Rearranging the equation into this form is a foundational step that simplifies the subsequent steps and makes the equation easier to solve. Without this step, it would be difficult to apply the standard solution techniques.
Step 2: Factoring the Quadratic Expression
Next, we attempt to factor the quadratic expression x² - 2x - 35. Factoring involves expressing the quadratic expression as a product of two binomials. We are looking for two numbers that multiply to c (-35) and add up to b (-2). Let's think about the factors of -35: (1, -35), (-1, 35), (5, -7), and (-5, 7). Among these pairs, 5 and -7 add up to -2. Therefore, we can rewrite the quadratic expression as:
(x + 5)(x - 7) = 0
Factoring is a powerful technique because it allows us to break down a complex expression into simpler components. Each binomial represents a potential solution to the equation. Factoring relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. In this case, either (x + 5) or (x - 7) must be equal to zero. Factoring is often the quickest and most efficient method for solving quadratic equations when the quadratic expression can be easily factored. However, not all quadratic equations can be factored easily, which is when other methods like completing the square or the quadratic formula become necessary.
Step 3: Solve for x
Now that we have factored the equation as (x + 5)(x - 7) = 0, we can set each factor equal to zero and solve for x:
- x + 5 = 0
Subtract 5 from both sides:
x = -5 - x - 7 = 0
Add 7 to both sides:
x = 7
Therefore, the solutions to the quadratic equation x² - 2x = 35 are x = -5 and x = 7. These are the two values of x that satisfy the original equation. Solving for x after factoring involves applying the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This step is crucial because it directly leads to the solutions of the equation. By setting each factor equal to zero and solving for x, we isolate the possible values of x that make the equation true. These values are the roots or zeros of the quadratic equation.
Step 4: Verify the Solutions
To ensure our solutions are correct, we can substitute each value of x back into the original equation x² - 2x = 35 and check if the equation holds true.
- For x = -5: (-5)² - 2(-5) = 25 + 10 = 35 (Correct)
- For x = 7: (7)² - 2(7) = 49 - 14 = 35 (Correct)
Both solutions satisfy the original equation. Verification is an essential step in the problem-solving process because it confirms that the solutions obtained are accurate. By substituting the solutions back into the original equation, we can check if the equation holds true. This step helps to catch any errors that may have occurred during the solving process. Verification provides confidence in the correctness of the solutions and ensures that the problem has been solved accurately. It is a good practice to always verify the solutions, especially in exams or when dealing with critical applications.
Alternative Method: Using the Quadratic Formula
Even though we successfully solved the equation by factoring, let's explore another method: the quadratic formula. The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² - 4ac)) / (2a)
In our equation x² - 2x - 35 = 0, we have a = 1, b = -2, and c = -35. Plugging these values into the quadratic formula, we get:
x = (2 ± √((-2)² - 4(1)(-35))) / (2(1))
x = (2 ± √(4 + 140)) / 2
x = (2 ± √144) / 2
x = (2 ± 12) / 2
This gives us two possible solutions:
- x = (2 + 12) / 2 = 14 / 2 = 7
- x = (2 - 12) / 2 = -10 / 2 = -5
As you can see, we arrive at the same solutions as before: x = -5 and x = 7. The quadratic formula is a powerful tool because it can be used to solve any quadratic equation, regardless of whether it can be factored easily. It is particularly useful when the quadratic expression is difficult or impossible to factor. The quadratic formula provides a systematic approach to finding the solutions, ensuring that all possible solutions are considered. While factoring is often quicker when it is possible, the quadratic formula is a reliable method that always works.
Conclusion
In summary, we've successfully solved the quadratic equation x² - 2x = 35 using factoring and verified our solutions. We also demonstrated how to solve the same equation using the quadratic formula. The solutions are x = -5 and x = 7. Understanding how to solve quadratic equations is a fundamental skill in mathematics with applications in various fields. Whether you prefer factoring, completing the square, or using the quadratic formula, mastering these techniques will enhance your problem-solving abilities and deepen your understanding of mathematical concepts.
For further exploration and practice on quadratic equations, you can visit Khan Academy for comprehensive lessons and exercises: https://www.khanacademy.org/.
