Solving X² - 14x + 45 = 0 Step-by-Step Guide

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Hey guys! Today, we're diving into the world of algebra to tackle a classic quadratic equation: x² - 14x + 45 = 0. Quadratic equations might seem intimidating at first, but trust me, once you understand the methods, they become quite manageable. This article will walk you through the process step-by-step, ensuring you grasp not just how to solve it, but why each step works. We’ll explore different methods, providing a robust understanding that will empower you to solve similar problems with confidence. Whether you're a student brushing up for an exam or just someone curious about math, this guide is for you. Let’s get started and unravel the mysteries of this equation together!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let’s take a moment to understand what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means it has the general form:

ax² + bx + c = 0

Where a, b, and c are constants, and x is the variable we're trying to find. The “quadratic” part comes from the fact that the highest power of x is 2 (x²). The coefficients a, b, and c play crucial roles in determining the nature and solutions of the equation. Understanding this basic form is the first step in mastering quadratic equations. Now, let's break down each component further:

  • a: This is the coefficient of the x² term. It's important because it determines the parabola's direction (whether it opens upwards or downwards) when the quadratic equation is graphed. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. Moreover, a cannot be zero, because if it were, the equation would become linear, not quadratic.
  • b: This is the coefficient of the x term. The b term influences the axis of symmetry of the parabola and its horizontal position on the graph. It works in conjunction with a to determine the vertex's x-coordinate, which is the point where the parabola changes direction.
  • c: This is the constant term, also known as the y-intercept. It’s the point where the parabola intersects the y-axis on a graph. The constant term c shifts the parabola vertically up or down the coordinate plane, relative to the origin.

In our equation, x² - 14x + 45 = 0, we can identify these coefficients as follows:

  • a = 1 (since the coefficient of x² is 1)
  • b = -14
  • c = 45

Recognizing these coefficients is crucial because they are used in various methods for solving quadratic equations, such as factoring, completing the square, and using the quadratic formula. With a clear understanding of the quadratic equation's components, we're now better equipped to explore the different solution methods. Each method offers a unique approach, and choosing the right one can make solving the equation much more efficient. Let’s move on to the first method: factoring.

Method 1: Factoring

Factoring is often the quickest way to solve a quadratic equation, especially when the roots are integers. The goal is to rewrite the quadratic expression as a product of two binomials. This method relies on finding two numbers that, when multiplied, give you the constant term (c) and when added, give you the coefficient of the x term (b). Let's see how this works for our equation, x² - 14x + 45 = 0.

Our equation has a = 1, b = -14, and c = 45. So, we need to find two numbers that multiply to 45 and add up to -14. To do this, we start by listing pairs of factors of 45:

  • 1 and 45
  • 3 and 15
  • 5 and 9

Now, let's consider the signs. Since the product is positive (+45) and the sum is negative (-14), both numbers must be negative. Looking at our list, -5 and -9 fit the bill perfectly:

  • (-5) * (-9) = 45
  • (-5) + (-9) = -14

Now that we’ve found our numbers, we can rewrite the quadratic equation in factored form. The factored form will look like this:

(x + first number)(x + second number) = 0

In our case, the factored form is:

(x - 5)(x - 9) = 0

This equation tells us that the product of two expressions, (x - 5) and (x - 9), equals zero. For a product to be zero, at least one of the factors must be zero. This principle is known as the Zero Product Property, and it's the key to finding the solutions (or roots) of the equation. So, we set each factor equal to zero and solve for x:

  1. x - 5 = 0 Add 5 to both sides: x = 5
  2. x - 9 = 0 Add 9 to both sides: x = 9

Therefore, the solutions to the quadratic equation x² - 14x + 45 = 0 are x = 5 and x = 9. These are the values of x that make the equation true. Factoring is an elegant and efficient method when it works, but not all quadratic equations can be easily factored. When factoring isn't straightforward, we need to turn to other methods, such as completing the square or using the quadratic formula. Let’s explore the next method: completing the square.

Method 2: Completing the Square

Completing the square is a powerful technique for solving quadratic equations, especially when they can't be easily factored. It involves manipulating the equation to create a perfect square trinomial on one side, which then allows us to solve for x by taking the square root. While it might seem a bit more complex than factoring, completing the square is a fundamental method that provides insight into the structure of quadratic equations. Let's apply this method to our equation, x² - 14x + 45 = 0.

The first step is to rearrange the equation so that the constant term (c) is on the right side of the equation. This isolates the x² and x terms on the left:

x² - 14x = -45

Now, we need to complete the square on the left side. This means we want to add a number to both sides of the equation that will make the left side a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be factored into the form (x + p)² or (x - p)². The number we need to add is determined by taking half of the coefficient of the x term (b), squaring it, and adding the result to both sides. In our case, b = -14, so we do the following:

  1. Take half of -14: -14 / 2 = -7
  2. Square the result: (-7)² = 49

So, we add 49 to both sides of the equation:

x² - 14x + 49 = -45 + 49

Now, the left side is a perfect square trinomial, which can be factored as (x - 7)²:

(x - 7)² = 4

Next, we take the square root of both sides of the equation. Remember to consider both the positive and negative square roots:

√(x - 7)² = ±√4

This simplifies to:

x - 7 = ±2

Now, we solve for x by adding 7 to both sides:

x = 7 ± 2

This gives us two possible solutions:

  1. x = 7 + 2 = 9
  2. x = 7 - 2 = 5

Thus, the solutions to the quadratic equation x² - 14x + 45 = 0 are x = 5 and x = 9, which are the same solutions we found by factoring. Completing the square is a versatile method because it works for any quadratic equation, regardless of whether it can be factored easily. It also provides a direct way to derive the quadratic formula, which is our next method. Understanding completing the square enhances our algebraic toolkit and sets the stage for the quadratic formula.

Method 3: Quadratic Formula

The quadratic formula is the ultimate tool for solving quadratic equations. It's a foolproof method that works for any quadratic equation, regardless of whether it can be factored or completed into a perfect square easily. It's derived from the process of completing the square, but it gives us a direct formula to plug in the coefficients and get the solutions. The quadratic formula is given by:

x = [-b ± √(b² - 4ac)] / 2a

Where a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant tells us about the nature of the solutions:

  • If b² - 4ac > 0, there are two distinct real solutions.
  • If b² - 4ac = 0, there is exactly one real solution (a repeated root).
  • If b² - 4ac < 0, there are no real solutions (two complex solutions).

Now, let’s apply the quadratic formula to our equation, x² - 14x + 45 = 0. As we identified earlier, a = 1, b = -14, and c = 45. Plugging these values into the formula, we get:

x = [-(-14) ± √((-14)² - 4 * 1 * 45)] / (2 * 1)

Let’s simplify this step by step:

  1. Simplify the numerator:

    • -(-14) = 14
    • (-14)² = 196
    • 4 * 1 * 45 = 180

    So, the equation becomes: x = [14 ± √(196 - 180)] / 2

  2. Continue simplifying inside the square root:

    • 196 - 180 = 16

    The equation is now: x = [14 ± √16] / 2

  3. Evaluate the square root:

    • √16 = 4

    The equation simplifies to: x = [14 ± 4] / 2

  4. Now, we find the two solutions by considering both the plus and minus signs:

    • x₁ = (14 + 4) / 2 = 18 / 2 = 9
    • x₂ = (14 - 4) / 2 = 10 / 2 = 5

So, the solutions to the quadratic equation x² - 14x + 45 = 0 are x = 5 and x = 9. Once again, we've arrived at the same solutions as with factoring and completing the square, but this time using the quadratic formula. The quadratic formula is invaluable because it's a universal method. No matter how complicated the coefficients are, or whether the solutions are integers, fractions, or irrational numbers, the quadratic formula will always provide the answer. This makes it an essential tool in your mathematical toolkit.

Choosing the Right Method

Now that we've explored three different methods for solving quadratic equations—factoring, completing the square, and the quadratic formula—you might be wondering: which method should I use? The answer depends on the specific equation you're dealing with and your personal preference. Each method has its strengths and weaknesses, and the best choice often comes down to efficiency and ease of use for a particular problem.

Factoring is generally the quickest method when it works. If you can easily identify two numbers that multiply to c and add up to b, factoring can save you a lot of time. However, not all quadratic equations can be factored easily, especially if the roots are not integers or simple fractions. Factoring is best suited for equations with integer coefficients and integer or simple fractional roots. When you encounter a quadratic equation, it's a good idea to first see if factoring is a viable option. Look for factor pairs of the constant term and check if they add up to the coefficient of the linear term. If you can find these factors quickly, you're in business. If not, it's time to consider other methods.

Completing the square is a more versatile method than factoring. It works for any quadratic equation, regardless of the nature of its roots. However, it can be more time-consuming and require more algebraic manipulation than factoring or using the quadratic formula directly. Completing the square is particularly useful when the coefficient of x² is 1 and the coefficient of x is an even number, as this simplifies the process of finding the number to add to both sides of the equation. Moreover, completing the square is a fundamental technique that provides a deeper understanding of quadratic equations. It’s the method used to derive the quadratic formula, so mastering it can give you a better conceptual grasp of quadratic relationships. While it may not always be the fastest method, it’s a valuable tool to have in your algebraic arsenal.

The quadratic formula is the most reliable and universally applicable method. It works for any quadratic equation, regardless of whether it can be factored or easily completed into a perfect square. While it might seem a bit intimidating at first due to its length, the quadratic formula is a direct plug-and-chug method once you know the coefficients a, b, and c. The downside of the quadratic formula is that it can sometimes involve more arithmetic calculations than the other methods, especially when dealing with large or complicated coefficients. However, its reliability makes it an indispensable tool. If you're unsure which method to use or if you've tried factoring and it's not working, the quadratic formula is your go-to solution. It's also particularly useful when the roots are irrational or complex, as these are not easily found by factoring or completing the square.

In summary, the choice of method depends on the specific equation and your comfort level with each technique. Factoring is quick when it works, completing the square is versatile and fundamental, and the quadratic formula is the ultimate fallback. With practice, you'll develop a sense of which method is best suited for different situations. The key is to be familiar with all three methods so you can choose the most efficient approach for any quadratic equation you encounter.

Conclusion

So there you have it, folks! We’ve successfully navigated the quadratic equation x² - 14x + 45 = 0 using three different methods: factoring, completing the square, and the quadratic formula. Each method gave us the same solutions, x = 5 and x = 9, but they approached the problem from different angles. Understanding these methods not only equips you to solve quadratic equations but also deepens your appreciation for the elegance and versatility of algebra.

We started by understanding what a quadratic equation is—a polynomial equation of the second degree with the general form ax² + bx + c = 0. Recognizing the coefficients a, b, and c is the first step in applying any solution method. We then dived into factoring, which is often the quickest method when the equation can be easily factored. By finding two numbers that multiply to the constant term and add up to the coefficient of the x term, we were able to rewrite the equation as a product of two binomials and solve for x using the Zero Product Property.

Next, we tackled completing the square, a method that involves transforming the equation into a perfect square trinomial. This technique is a bit more involved but works for any quadratic equation. By adding a carefully chosen number to both sides of the equation, we created a perfect square on one side, took the square root, and solved for x. Completing the square is not only a powerful problem-solving tool but also provides a deeper understanding of the structure of quadratic equations.

Finally, we unleashed the power of the quadratic formula, the ultimate solution for any quadratic equation. This formula, x = [-b ± √(b² - 4ac)] / 2a, is derived from completing the square and provides a direct way to find the solutions by plugging in the coefficients a, b, and c. The quadratic formula is particularly useful when factoring is difficult or impossible, and it works for all types of roots, including real, irrational, and complex numbers.

Choosing the right method depends on the specific equation and your comfort level. Factoring is efficient when it works, completing the square is versatile and fundamental, and the quadratic formula is the reliable fallback. By mastering all three methods, you become a more confident and capable problem solver.

Remember, practice makes perfect. The more you work with quadratic equations, the more comfortable and proficient you’ll become. So, keep solving, keep exploring, and keep enjoying the beauty of mathematics!