Solving X² = 45: A Step-by-Step Guide

Understanding how to solve quadratic equations is a fundamental skill in mathematics. In this comprehensive guide, we'll break down the process of solving the equation x² = 45 for real number solutions. We'll cover each step in detail, ensuring you understand the underlying concepts and can confidently tackle similar problems. Whether you're a student looking to improve your algebra skills or simply curious about mathematical problem-solving, this article is for you. Let's dive in and explore how to find the solutions, simplify them, and express them correctly.

Understanding the Equation x² = 45

When dealing with the equation x² = 45, it's essential to understand the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In our case, the equation x² = 45 can be rewritten as x² - 45 = 0, which fits the general form with a = 1, b = 0, and c = -45. Solving this equation means finding the values of x that satisfy the equation, or in other words, make the equation true.

The equation x² = 45 presents a situation where a number, when multiplied by itself, equals 45. This naturally leads us to the concept of square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. However, it's crucial to remember that every positive number has two square roots: a positive square root and a negative square root. This is because both the positive and negative values, when squared, result in the same positive number. For instance, both 3 and -3 are square roots of 9 because 3² = 9 and (-3)² = 9. This principle is particularly important when solving equations like x² = 45, as we need to consider both the positive and negative square roots to find all possible solutions.

Considering the properties of square roots, the initial approach to solving x² = 45 involves isolating x by taking the square root of both sides of the equation. This step is mathematically sound and follows the fundamental principle of maintaining equality in an equation. Whatever operation we perform on one side of the equation, we must also perform on the other side to preserve the balance. This approach directly addresses the question of what number, when squared, equals 45. By applying the square root operation, we begin to unravel the equation and reveal the potential values of x that satisfy the given condition. However, it's crucial to remember that taking the square root introduces both positive and negative solutions, as squaring either a positive or a negative number yields a positive result. Therefore, we must consider both possibilities when proceeding with the solution.

Step-by-Step Solution for x² = 45

Now, let's delve into the step-by-step process of solving the equation x² = 45. This will provide a clear and methodical approach to finding the solutions.

Step 1: Take the Square Root of Both Sides

The first step in solving x² = 45 is to take the square root of both sides of the equation. This operation is crucial because it helps us isolate x. By doing so, we are essentially reversing the squaring operation. When we take the square root of x², we get |x|, which represents the absolute value of x. On the other side, we get ±√45, representing both the positive and negative square roots of 45. Remember, when solving equations involving squares, it's essential to consider both positive and negative roots because both can satisfy the equation. So, the equation now looks like this:

|x| = ±√45

Step 2: Simplify the Square Root

The next step involves simplifying the square root of 45. To do this, we need to find the prime factorization of 45. The prime factorization of 45 is 3 × 3 × 5, which can also be written as 3² × 5. Now, we can rewrite √45 as √(3² × 5). Using the property of square roots that √(a × b) = √a × √b, we can separate this into √3² × √5. The square root of 3² is simply 3, so we have 3√5. This simplification makes the solution much cleaner and easier to work with. Therefore, our equation now becomes:

x = ±3√5

Step 3: Express the Solutions

Now that we've simplified the square root, we can express the solutions for x. Since we have both a positive and a negative root, we have two solutions: x = 3√5 and x = -3√5. These are the two real numbers that, when squared, give us 45. It's important to present both solutions to ensure a complete and accurate answer. These solutions can be expressed separately or combined using the ± notation, which concisely represents both values. Thus, the final solutions to the equation x² = 45 are:

x = 3√5, -3√5

Alternative Methods for Solving Quadratic Equations

While taking the square root is a direct method for solving equations in the form x² = c, where c is a constant, there are other methods applicable to more complex quadratic equations. Understanding these methods provides a broader toolkit for solving various types of quadratic equations. Let's explore some alternative approaches.

Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic expression as a product of two binomials. This method is particularly effective when the quadratic equation can be easily factored. For example, consider the equation x² + 5x + 6 = 0. To solve this by factoring, we need to find two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the x term). These numbers are 2 and 3. So, we can rewrite the equation as (x + 2)(x + 3) = 0. Setting each factor equal to zero gives us the solutions x = -2 and x = -3. Factoring is a straightforward method, but it is most effective when the quadratic equation has integer solutions and can be easily factored. It relies on recognizing patterns and relationships between the coefficients and constants in the equation. However, not all quadratic equations can be easily factored, making it necessary to have other methods in your problem-solving arsenal.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation of the form ax² + bx + c = 0. This formula is derived from the method of completing the square and provides a direct way to find the solutions, regardless of whether the equation can be factored or not. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

To use the quadratic formula, you simply need to identify the coefficients a, b, and c from the quadratic equation and plug them into the formula. For example, consider the equation 2x² + 5x - 3 = 0. Here, a = 2, b = 5, and c = -3. Substituting these values into the formula, we get:

x = (-5 ± √(5² - 4(2)(-3))) / (2(2))

Simplifying this expression, we find the solutions for x. The quadratic formula is a powerful tool because it works for all quadratic equations, including those with real and complex solutions. The discriminant (b² - 4ac) within the formula determines the nature of the solutions. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, there are two complex solutions. The quadratic formula is an indispensable method in algebra, ensuring you can solve any quadratic equation, no matter how complex.

Completing the Square

Completing the square is a method that transforms a quadratic equation into a perfect square trinomial, which can then be easily solved. This method involves manipulating the equation to create a square on one side, making it simpler to find the solutions. The general process involves adding and subtracting a specific value to both sides of the equation to complete the square. For example, consider the equation x² + 6x + 5 = 0. To complete the square, we first move the constant term to the right side: x² + 6x = -5. Next, we take half of the coefficient of the x term (which is 6), square it (which is (6/2)² = 9), and add it to both sides: x² + 6x + 9 = -5 + 9. This transforms the left side into a perfect square trinomial: (x + 3)² = 4. Now, we can take the square root of both sides: x + 3 = ±2. Solving for x gives us the solutions x = -1 and x = -5. Completing the square is particularly useful for understanding the structure of quadratic equations and deriving the quadratic formula. It provides a deeper insight into the properties of quadratic equations and is a valuable technique for solving them, especially when factoring is not straightforward. This method also highlights the relationship between the algebraic form of the equation and its geometric representation, making it a powerful tool in mathematical problem-solving.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can help illustrate the practical significance of solving quadratic equations. From physics to engineering to finance, quadratic equations play a crucial role in modeling and solving problems.

Physics

In physics, quadratic equations are frequently used to describe projectile motion. When an object is thrown or launched into the air, its trajectory can be modeled using a quadratic equation. For example, the height of a projectile as a function of time can be represented by the equation h(t) = -16t² + vt + h₀, where h(t) is the height at time t, v is the initial vertical velocity, and h₀ is the initial height. The -16t² term accounts for the effect of gravity. By setting h(t) to zero and solving the quadratic equation, we can find the time it takes for the projectile to hit the ground. This type of calculation is essential in fields like sports, where understanding the trajectory of a ball is critical, and in engineering, where designing projectiles or understanding ballistic motion is necessary. Quadratic equations help physicists predict and analyze the motion of objects under the influence of gravity, providing valuable insights into the physical world. These equations also help in designing various instruments and technologies that rely on projectile motion, such as cannons and missiles.

Engineering

Engineers use quadratic equations in various applications, including designing bridges, buildings, and other structures. Quadratic equations are essential in structural analysis, helping engineers determine the stresses and strains on different parts of a structure. For instance, the shape of a parabolic arch, which is a common design element in bridges, is described by a quadratic equation. The equation helps engineers calculate the optimal curvature for the arch to ensure it can support the intended load. In electrical engineering, quadratic equations are used to analyze circuits and calculate power dissipation. They are also used in control systems engineering to model the behavior of systems and design controllers that maintain stability and performance. Quadratic equations provide a mathematical framework for understanding and designing systems and structures, ensuring they are safe, efficient, and reliable. The ability to solve and manipulate these equations is a fundamental skill for engineers across various disciplines, enabling them to address complex problems and create innovative solutions.

Finance

In finance, quadratic equations are used to model various financial scenarios, such as calculating loan payments, determining investment returns, and assessing risk. For example, the future value of an investment with compound interest can be modeled using a quadratic equation. Understanding these models helps financial analysts and investors make informed decisions. Quadratic equations can also be used in option pricing models, where the value of an option depends on several factors that can be related through a quadratic equation. These models help traders and investors assess the potential profit and risk associated with different investment strategies. Quadratic equations provide a mathematical basis for understanding and managing financial risks and returns, making them an invaluable tool in the financial industry. The ability to apply these equations in financial analysis allows professionals to develop strategies that maximize returns while minimizing risk, contributing to better financial planning and decision-making.

Conclusion

In this comprehensive guide, we've explored how to solve the equation x² = 45 step by step, emphasizing the importance of understanding both positive and negative square roots. We've also delved into alternative methods for solving quadratic equations, such as factoring, the quadratic formula, and completing the square. Furthermore, we've highlighted the numerous real-world applications of quadratic equations in fields like physics, engineering, and finance. By mastering the techniques for solving quadratic equations, you'll be well-equipped to tackle a wide range of mathematical and practical problems. Remember, practice is key, so continue to apply these methods to different equations to strengthen your understanding and skills. Understanding the principles behind these methods will empower you to approach any quadratic equation with confidence and precision.

For further exploration of quadratic equations and related topics, consider visiting Khan Academy's algebra section.