Solving The Quadratic Equation X² + 10x + 25 = 36 A Comprehensive Guide

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Deciphering Quadratic Equations: Navigating the realm of algebra often leads us to quadratic equations, mathematical expressions of the form ax² + bx + c = 0. These equations, characterized by their squared variable term, hold significant importance in various fields, from physics and engineering to economics and computer science. The equation at hand, x² + 10x + 25 = 36, presents a compelling challenge, inviting us to unravel its hidden solutions.

Method 1: Embracing the Square Root Property

To embark on our quest for solutions, we can employ the square root property, a powerful tool that allows us to isolate the variable x. This method hinges on the principle that if two expressions are equal, their square roots must also be equal. Applying this principle to our equation, we begin by recognizing that the left-hand side, x² + 10x + 25, is a perfect square trinomial. This means it can be factored into the form (x + 5)².

Now, our equation transforms into (x + 5)² = 36. Taking the square root of both sides, we arrive at x + 5 = ±6. It's crucial to remember the ± sign, as both positive and negative square roots satisfy the equation. This step unveils two potential paths: x + 5 = 6 and x + 5 = -6.

Solving for x in each case, we find x = 1 and x = -11. These are the solutions to our quadratic equation, the values of x that make the equation true. We can verify these solutions by plugging them back into the original equation and confirming that both sides are equal.

Step-by-step Breakdown:

  1. Recognize the perfect square trinomial: x² + 10x + 25 = (x + 5)²
  2. Rewrite the equation: (x + 5)² = 36
  3. Take the square root of both sides: x + 5 = ±6
  4. Solve for x: x = 1 and x = -11

Method 2: The Art of Rearranging and Factoring

An alternative approach involves rearranging the equation and employing the technique of factoring. To begin, we subtract 36 from both sides of the equation, setting it equal to zero: x² + 10x + 25 - 36 = 0. This simplifies to x² + 10x - 11 = 0.

Now, we seek to factor the quadratic expression on the left-hand side. Factoring involves expressing the quadratic as a product of two binomials. In this case, we look for two numbers that multiply to -11 and add up to 10. These numbers are 11 and -1. Thus, we can factor the expression as (x + 11)(x - 1) = 0.

The zero-product property comes into play here, stating that if the product of two factors is zero, then at least one of the factors must be zero. This leads us to two possibilities: x + 11 = 0 and x - 1 = 0. Solving these equations, we again find the solutions x = -11 and x = 1.

Step-by-step Breakdown:

  1. Subtract 36 from both sides: x² + 10x - 11 = 0
  2. Factor the quadratic expression: (x + 11)(x - 1) = 0
  3. Apply the zero-product property: x + 11 = 0 or x - 1 = 0
  4. Solve for x: x = -11 and x = 1

Method 3: Unleashing the Quadratic Formula

For a more general approach applicable to any quadratic equation, we can turn to the quadratic formula. This formula provides a direct solution for x in the equation ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a. In our case, a = 1, b = 10, and c = -11.

Plugging these values into the formula, we get x = (-10 ± √(10² - 4 * 1 * -11)) / (2 * 1). This simplifies to x = (-10 ± √(100 + 44)) / 2, and further to x = (-10 ± √144) / 2. Since √144 = 12, we have x = (-10 ± 12) / 2.

This yields two solutions: x = (-10 + 12) / 2 = 1 and x = (-10 - 12) / 2 = -11, consistent with our previous findings.

Step-by-step Breakdown:

  1. Identify a, b, and c: a = 1, b = 10, c = -11
  2. Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
  3. Substitute values: x = (-10 ± √(10² - 4 * 1 * -11)) / (2 * 1)
  4. Simplify: x = (-10 ± √144) / 2
  5. Solve for x: x = 1 and x = -11

Conclusion: The Solutions Unveiled

Through three distinct methods – the square root property, factoring, and the quadratic formula – we have arrived at the same solutions for the equation x² + 10x + 25 = 36: x = 1 and x = -11. These solutions represent the values of x that satisfy the equation, the points where the parabola represented by the quadratic expression intersects the horizontal line y = 36. The journey of solving this quadratic equation has not only revealed the answers but has also illuminated the diverse techniques available in the realm of algebra.

Understanding the Significance of Solutions: The solutions to a quadratic equation, often called roots or zeros, hold profound significance in various mathematical and real-world contexts. They represent the points where the graph of the quadratic function intersects the x-axis, providing valuable insights into the function's behavior and properties. In practical applications, these solutions can model physical phenomena, predict outcomes, and optimize processes. The solutions x = 1 and x = -11, which we diligently uncovered, are more than just numerical answers; they are keys to unlocking the hidden meaning and potential of the equation x² + 10x + 25 = 36.

The Possible Solutions for x in the Equation x² + 10x + 25 = 36

When confronted with the equation x² + 10x + 25 = 36, our goal is to determine the possible values of x that satisfy this equation. This type of equation is a quadratic equation, and there are several methods we can use to find its solutions. We will explore these methods to ensure a comprehensive understanding of how to solve this type of problem.

Method 1: Solving by Factoring

One effective method is to solve the quadratic equation by factoring. To do this, we first need to rewrite the equation in the standard form, which is ax² + bx + c = 0. In our case, we need to subtract 36 from both sides of the equation to set it to zero:

x² + 10x + 25 - 36 = 0

This simplifies to:

x² + 10x - 11 = 0

Now, we look for two numbers that multiply to -11 (the constant term) and add up to 10 (the coefficient of the x term). The numbers that satisfy these conditions are 11 and -1. Thus, we can factor the quadratic expression as follows:

(x + 11)(x - 1) = 0

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

  1. x + 11 = 0 implies x = -11
  2. x - 1 = 0 implies x = 1

So, the solutions for x are -11 and 1. These are the values that, when substituted into the original equation, make the equation true.

Understanding Factoring: Factoring is a crucial technique in algebra that involves expressing a quadratic expression as a product of two binomials. This method is particularly efficient when the quadratic equation has integer solutions. In our case, the ability to factor the expression (x² + 10x - 11) into (x + 11)(x - 1) allowed us to quickly identify the roots of the equation. Factoring not only simplifies the process of solving quadratic equations but also provides insights into the structure and properties of the quadratic function itself. Mastering factoring techniques is essential for tackling more complex algebraic problems and understanding the relationships between different mathematical concepts.

Method 2: Using the Quadratic Formula

Another powerful method to solve quadratic equations is by using the quadratic formula. The quadratic formula is a general formula that provides the solutions for any quadratic equation in the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / (2a)

First, we need to ensure the equation is in the standard form. As we found earlier, the equation can be rewritten as:

x² + 10x - 11 = 0

In this equation:

  • a = 1* (the coefficient of x²)
  • b = 10* (the coefficient of x)
  • c = -11* (the constant term)

Now, we substitute these values into the quadratic formula:

x = [-10 ± √(10² - 4 * 1 * -11)] / (2 * 1)

Simplifying the expression under the square root:

x = [-10 ± √(100 + 44)] / 2

x = [-10 ± √144] / 2

Since √144 = 12, we have:

x = [-10 ± 12] / 2

This gives us two possible solutions:

  1. x = (-10 + 12) / 2 = 2 / 2 = 1
  2. x = (-10 - 12) / 2 = -22 / 2 = -11

Thus, the solutions for x are again -11 and 1. This method is particularly useful for equations that are difficult or impossible to factor.

Unveiling the Power of the Quadratic Formula: The quadratic formula stands as a cornerstone in algebra, offering a universal solution to quadratic equations, irrespective of their factorability. This formula encapsulates a fundamental mathematical principle, allowing us to navigate the complexities of quadratic expressions with precision and confidence. The beauty of the quadratic formula lies not only in its ability to provide solutions but also in its capacity to reveal the nature of those solutions – whether they are real or complex, rational or irrational. By understanding the discriminant (b² - 4ac) within the formula, we gain deeper insights into the characteristics of the quadratic equation and its corresponding graph. The quadratic formula is more than just a tool; it is a gateway to a profound understanding of quadratic functions and their applications in various fields, from physics and engineering to economics and computer science.

Method 3: Completing the Square

Completing the square is another technique to solve quadratic equations. This method involves transforming the quadratic equation into a perfect square trinomial, making it easier to solve. Starting with the original equation:

x² + 10x + 25 = 36

Notice that the left side of the equation is already a perfect square trinomial. It can be factored as:

(x + 5)² = 36

Now, we take the square root of both sides of the equation:

√(x + 5)² = ±√36

This simplifies to:

x + 5 = ±6

We now have two separate equations to solve:

  1. x + 5 = 6 implies x = 6 - 5 = 1
  2. x + 5 = -6 implies x = -6 - 5 = -11

Once again, the solutions for x are -11 and 1. This method is especially useful when the quadratic equation is already in or can be easily transformed into a perfect square form.

Completing the Square: A Masterful Technique: Completing the square is not merely a method for solving quadratic equations; it is an art form that unveils the underlying structure of quadratic expressions. This technique involves transforming a quadratic expression into a perfect square trinomial, a form that reveals the vertex of the corresponding parabola and provides valuable insights into the function's behavior. Completing the square allows us to rewrite quadratic equations in a way that makes them easier to solve, and it also serves as a foundation for deriving the quadratic formula itself. This method showcases the elegance and power of algebraic manipulation, demonstrating how seemingly complex equations can be simplified through careful and strategic transformations. Mastering the technique of completing the square not only enhances problem-solving skills but also fosters a deeper appreciation for the interconnectedness of mathematical concepts.

The Solutions

Therefore, the possible solutions for x in the equation x² + 10x + 25 = 36 are -11 and 1. These solutions can be verified by substituting them back into the original equation and confirming that the equation holds true. We have employed three different methods to demonstrate that these are indeed the correct solutions, ensuring a thorough and comprehensive understanding of the problem.

Confirming the Solutions: The solutions we have derived through factoring, the quadratic formula, and completing the square consistently point to x = -11 and x = 1 as the values that satisfy the equation x² + 10x + 25 = 36. To solidify our understanding, it is crucial to verify these solutions by substituting them back into the original equation. This process of verification not only confirms the accuracy of our calculations but also reinforces the fundamental principle that the solutions to an equation are the values that make the equation true. By plugging in x = -11 and x = 1, we can witness firsthand how the equation balances, affirming the validity of our findings. This step is a testament to the importance of precision and attention to detail in mathematical problem-solving.

Answering the Question: Possible Solutions for x

Having explored three different methodologies—factoring, the quadratic formula, and completing the square—we've consistently arrived at the same solutions for the equation x² + 10x + 25 = 36. These solutions are x = -11 and x = 1.

Thus, the possible solutions for x are -11 and 1.