Solving For X In The Equation X^2 - 12x + 36 = 90 A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of solving the quadratic equation x^2 - 12x + 36 = 90. Quadratic equations are fundamental in mathematics and frequently appear in various fields, including physics, engineering, and computer science. Understanding how to solve them is crucial for anyone pursuing studies or careers in these areas. This article will provide a step-by-step approach to finding the value(s) of x that satisfy the given equation, ensuring clarity and thoroughness throughout the explanation. By the end of this guide, you will not only understand the solution but also grasp the underlying principles and techniques applicable to solving similar problems.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are also known as roots or zeros. These roots represent the values of x that make the equation true. Solving quadratic equations is a fundamental skill in algebra, and there are several methods to achieve this, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of quadratic equations. For instance, factoring is efficient when the equation can be easily factored, while the quadratic formula provides a universal approach that works for any quadratic equation. Understanding these methods and when to apply them is crucial for mastering quadratic equations.

Methods for Solving Quadratic Equations

There are three primary methods for solving quadratic equations:

  1. Factoring: This method involves expressing the quadratic equation as a product of two binomials. If the equation can be factored easily, this method is often the quickest and most straightforward. Factoring relies on finding two numbers that multiply to give the constant term c and add up to the coefficient b in the equation ax^2 + bx + c = 0. Once the equation is factored, each factor is set to zero, leading to the solutions for x. However, factoring is not always feasible, especially when the roots are not rational numbers.

  2. Completing the Square: This method involves transforming the quadratic equation into a perfect square trinomial. Completing the square is a powerful technique that can be used to solve any quadratic equation. It involves manipulating the equation to form a square on one side, which can then be easily solved by taking the square root. This method is particularly useful when the equation cannot be easily factored or when deriving the quadratic formula. The process of completing the square involves adding and subtracting a specific value to both sides of the equation to create a perfect square trinomial.

  3. Quadratic Formula: This formula provides a direct way to find the solutions of any quadratic equation. The quadratic formula is derived from the method of completing the square and is given by: x = (-b ± √(b^2 - 4ac)) / (2a). This formula is applicable to all quadratic equations, regardless of whether they can be factored or not. The quadratic formula is a universal tool in solving quadratic equations and is especially useful when the coefficients are complex or when high precision is required.

Step-by-Step Solution for x^2 - 12x + 36 = 90

Now, let's solve the given equation x^2 - 12x + 36 = 90 step by step:

Step 1: Simplify the Equation

The initial step involves simplifying the equation to bring it into the standard quadratic form (ax^2 + bx + c = 0). To do this, subtract 90 from both sides of the equation:

  • x^2 - 12x + 36 - 90 = 0
  • x^2 - 12x - 54 = 0

Step 2: Choose a Solution Method

We can solve this quadratic equation using either completing the square or the quadratic formula. Factoring is not straightforward in this case, so we will opt for completing the square. The coefficients are a = 1, b = -12, and c = -54. Completing the square involves transforming the equation into a perfect square trinomial, which can then be easily solved.

Step 3: Complete the Square

To complete the square, we need to add and subtract (b/2)^2 from the left side of the equation. In this case, b = -12, so (b/2)^2 = (-12/2)^2 = (-6)^2 = 36. Notice that the left side of our equation already contains x^2 - 12x + 36, which is a perfect square trinomial. Thus, we can rewrite the equation as:

  • (x - 6)^2 - 54 = 0

Now, we add 54 to both sides to isolate the squared term:

  • (x - 6)^2 = 54

Step 4: Solve for x

To solve for x, take the square root of both sides of the equation:

  • √(x - 6)^2 = ±√54
  • x - 6 = ±√54

Simplify the square root of 54. Since 54 = 9 * 6, we have √54 = √(9 * 6) = √9 * √6 = 3√6. Thus,

  • x - 6 = ±3√6

Now, add 6 to both sides to isolate x:

  • x = 6 ± 3√6

Alternative Method: Using the Quadratic Formula

Alternatively, we can use the quadratic formula to solve the equation x^2 - 12x - 54 = 0. The quadratic formula is given by:

  • x = (-b ± √(b^2 - 4ac)) / (2a)

In our equation, a = 1, b = -12, and c = -54. Substitute these values into the formula:

  • x = (-(-12) ± √((-12)^2 - 4(1)(-54))) / (2(1))
  • x = (12 ± √(144 + 216)) / 2
  • x = (12 ± √360) / 2

Simplify the square root of 360. Since 360 = 36 * 10, we have √360 = √(36 * 10) = √36 * √10 = 6√10. Thus,

  • x = (12 ± 6√10) / 2

Now, divide both terms in the numerator by 2:

  • x = 6 ± 3√10

Conclusion

Therefore, the solutions for the equation x^2 - 12x + 36 = 90 are x = 6 ± 3√10. We have demonstrated two methods for solving this quadratic equation: completing the square and using the quadratic formula. Both methods yield the same solutions, providing a robust understanding of how to tackle quadratic equations. Mastering these techniques is essential for anyone dealing with mathematical problems in various fields. By understanding these methods, you are well-equipped to solve a wide range of quadratic equations effectively.

Final Answer: A. x = 6 ± 3√10

In conclusion, the correct answer to the equation x^2 - 12x + 36 = 90 is A. x = 6 ± 3√10. This comprehensive guide has walked you through the step-by-step process of solving the equation using both completing the square and the quadratic formula. By understanding these methods, you can confidently approach similar problems and enhance your problem-solving skills in mathematics.