Solving (x-7)^2=36 Step-by-Step Guide To Quadratic Equations

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In the realm of mathematics, quadratic equations hold a prominent position, often encountered in various fields ranging from physics to engineering. These equations, characterized by the presence of a squared variable, present a unique challenge in finding their solutions. One common form of a quadratic equation is (x - a)^2 = b, where 'a' and 'b' are constants. To solve such equations, we embark on a journey of algebraic manipulation, employing techniques like taking square roots and isolating the variable 'x'. This article delves into the intricacies of solving the equation (x - 7)^2 = 36, meticulously outlining each step involved in the solution process. By understanding the underlying principles and applying the appropriate methods, we can effectively decipher the values of 'x' that satisfy this equation.

At the heart of our endeavor lies the understanding of quadratic equations. These equations, distinguished by their squared variable, often possess two distinct solutions. This duality arises from the nature of squaring a number, where both positive and negative values can yield the same result. For instance, both 3 squared (3^2) and -3 squared (-3)^2 equal 9. This fundamental concept plays a crucial role in solving equations like (x - 7)^2 = 36. To unravel the solutions, we must recognize that the expression (x - 7) can potentially be either the positive or negative square root of 36. This realization forms the cornerstone of our solution strategy, guiding us towards identifying the two possible values of 'x' that satisfy the equation. By acknowledging the inherent duality of quadratic equations, we pave the way for a comprehensive and accurate solution.

To embark on the journey of solving the equation (x - 7)^2 = 36, we must carefully navigate the steps of algebraic manipulation. Our initial move involves taking the square root of both sides of the equation. This seemingly simple act is pivotal, as it unravels the squared term, paving the way for isolating 'x'. Upon taking the square root, we encounter a crucial juncture: acknowledging both the positive and negative roots of 36. This recognition stems from the fundamental principle that both 6 and -6, when squared, result in 36. Consequently, we arrive at two distinct equations: x - 7 = 6 and x - 7 = -6. These equations represent the two possible scenarios arising from the square root operation. To solve for 'x' in each scenario, we employ the technique of adding 7 to both sides of each equation. This isolates 'x', revealing the two solutions: x = 13 and x = 1. These solutions, derived through meticulous algebraic manipulation, represent the values of 'x' that satisfy the original equation. By understanding and applying these steps, we successfully navigate the intricacies of solving quadratic equations.

Detailed Breakdown of the Solution Process

  1. Taking the square root of both sides: The initial step in solving the equation (x - 7)^2 = 36 involves taking the square root of both sides. This operation aims to eliminate the squared term, bringing us closer to isolating 'x'. By applying the square root, we transform the equation into √(x - 7)^2 = ±√36. The ± symbol is crucial here, as it signifies that we must consider both the positive and negative square roots of 36.

  2. Considering both positive and negative roots: The square root of 36 yields two possible values: 6 and -6. This is because both 6 squared (6^2) and -6 squared (-6)^2 equal 36. Consequently, we obtain two equations: x - 7 = 6 and x - 7 = -6. These equations represent the two possible scenarios arising from the square root operation. Failing to consider both roots would lead to an incomplete solution, missing one of the values of 'x' that satisfies the original equation.

  3. Solving for x in each case: To solve for 'x' in each equation, we employ the technique of adding 7 to both sides. For the equation x - 7 = 6, adding 7 to both sides yields x = 13. Similarly, for the equation x - 7 = -6, adding 7 to both sides gives us x = 1. These two values, x = 13 and x = 1, represent the solutions to the original equation. They are the values of 'x' that, when substituted into the equation (x - 7)^2 = 36, make the equation true.

Having obtained the solutions x = 13 and x = 1, it is paramount to verify their accuracy. This process involves substituting each solution back into the original equation, (x - 7)^2 = 36, to ensure that the equation holds true. When we substitute x = 13, we get (13 - 7)^2 = 6^2 = 36, which confirms that x = 13 is indeed a solution. Similarly, when we substitute x = 1, we get (1 - 7)^2 = (-6)^2 = 36, further solidifying the validity of our solutions. This verification step is essential in mathematical problem-solving, as it safeguards against potential errors and instills confidence in the correctness of the answers. By meticulously verifying our solutions, we ensure the integrity of our mathematical endeavors.

Substituting x = 13 into the equation:

Substituting x = 13 into the original equation, (x - 7)^2 = 36, we get:

(13 - 7)^2 = 36

Simplifying the expression inside the parentheses, we have:

(6)^2 = 36

Evaluating the square, we obtain:

36 = 36

This result confirms that x = 13 is a valid solution, as it satisfies the original equation.

Substituting x = 1 into the equation:

Substituting x = 1 into the original equation, (x - 7)^2 = 36, we get:

(1 - 7)^2 = 36

Simplifying the expression inside the parentheses, we have:

(-6)^2 = 36

Evaluating the square, we obtain:

36 = 36

This result further confirms that x = 1 is also a valid solution, as it satisfies the original equation.

In conclusion, we have successfully navigated the intricacies of solving the quadratic equation (x - 7)^2 = 36. Through a meticulous step-by-step approach, we have unveiled the two solutions: x = 13 and x = 1. This journey has underscored the importance of understanding the fundamental principles of quadratic equations, particularly the concept of both positive and negative roots. By taking the square root of both sides of the equation, we opened the door to two distinct possibilities, each leading to a unique solution. Furthermore, we emphasized the significance of verification, ensuring the accuracy of our solutions by substituting them back into the original equation. This process not only validates our answers but also reinforces our understanding of the underlying mathematical concepts. The ability to solve quadratic equations is a valuable asset in various fields, from mathematics and physics to engineering and computer science. By mastering this skill, we empower ourselves to tackle a wide range of problems and challenges.