Determine The Solution For X2 36 12x

Solving the Quadratic Equation: x² + 12x - 36 = 0

Determining the solution for the quadratic equation x² + 12x - 36 = 0 involves finding the values of 'x' that satisfy the equation. Quadratic equations, characterized by their highest power of x being 2, can be solved using several methods. This article will explore various approaches to solve this specific equation, explaining the underlying principles and showcasing the steps involved. We'll delve into factoring, completing the square, and using the quadratic formula, providing a comprehensive understanding of this fundamental algebraic concept.

Understanding Quadratic Equations

Before diving into the solution, let's briefly review the general form of a quadratic equation: ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our equation, x² + 12x - 36 = 0, fits this form with a = 1, b = 12, and c = -36. The solutions, also known as roots or zeros, represent the x-intercepts of the parabola represented by the equation when graphed on a coordinate plane.

Method 1: Factoring

Factoring is a method of solving quadratic equations by expressing the equation as a product of two linear expressions. This method is often the quickest if the equation can be easily factored. We look for two numbers that add up to 'b' (12) and multiply to 'c' (-36). Let's find these numbers:

• Finding the factors: We need two numbers that add to 12 and multiply to -36. After some trial and error (or using the AC method if you're familiar with it), we find that 18 and -6 fulfill this condition (18 + (-6) = 12 and 18 * (-6) = -36).

• Factoring the equation: Using these numbers, we can rewrite the equation as: (x + 18)(x - 6) = 0

• Solving for x: This equation is true if either (x + 18) = 0 or (x - 6) = 0. Solving these individual linear equations gives us two solutions:

  • x + 18 = 0 => x = -18
  • x - 6 = 0 => x = 6

Therefore, the solutions to the equation x² + 12x - 36 = 0 using the factoring method are x = -18 and x = 6.

Method 2: Completing the Square

Completing the square is a more general method that works for all quadratic equations, even those that are not easily factorable. This method involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

• Move the constant term: First, move the constant term (-36) to the right side of the equation: x² + 12x = 36

• Find the value to complete the square: Take half of the coefficient of the x term (12/2 = 6), square it (6² = 36), and add it to both sides of the equation: x² + 12x + 36 = 36 + 36

• Factor the perfect square trinomial: The left side is now a perfect square trinomial: (x + 6)² = 72

• Solve for x: Take the square root of both sides: x + 6 = ±√72

• Simplify and solve: Simplify √72 (which is 6√2) and solve for x:

  • x + 6 = 6√2 => x = -6 + 6√2
  • x + 6 = -6√2 => x = -6 - 6√2

Thus, the solutions using the completing the square method are x = -6 + 6√2 and x = -6 - 6√2. Notice that these are the exact solutions; the factoring method yielded simpler, integer solutions in this specific case.

Method 3: The Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. It provides the solutions directly, regardless of whether the equation is easily factorable. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

where a, b, and c are the coefficients from the standard quadratic equation ax² + bx + c = 0.

• Substitute the values: Substitute the values a = 1, b = 12, and c = -36 into the quadratic formula:

x = [-12 ± √(12² - 4 * 1 * -36)] / (2 * 1)

• Simplify:

x = [-12 ± √(144 + 144)] / 2 x = [-12 ± √288] / 2 x = [-12 ± 12√2] / 2 x = -6 ± 6√2

• Solutions: This gives us the same solutions as the completing the square method: x = -6 + 6√2 and x = -6 - 6√2.

Comparing the Methods

All three methods—factoring, completing the square, and the quadratic formula—yield the same solutions. However, the choice of method depends on the specific equation:

• Factoring: Easiest and quickest if the equation is easily factorable. It provides simple, integer solutions if they exist.

• Completing the square: A more general method applicable to all quadratic equations, providing exact solutions. It can be more time-consuming than factoring.

• Quadratic formula: A universal method applicable to all quadratic equations. It always provides the solutions, though the solutions might be irrational numbers. It's particularly useful when factoring is difficult or impossible.

Understanding the Solutions in Context

The solutions x = -18 and x = 6 (from the factoring method) and x = -6 ± 6√2 (from completing the square and the quadratic formula) represent the points where the parabola y = x² + 12x - 36 intersects the x-axis. These are the x-values for which y = 0. The different forms of the solutions simply reflect different ways of expressing the same mathematical concepts. The approximate values of the irrational solutions are x ≈ 2.485 and x ≈ -14.485.

Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts. They find wide applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains in materials.
• Economics: Modeling supply and demand, determining optimal production levels.
• Computer graphics: Creating curves and shapes in computer-aided design.

Understanding how to solve quadratic equations is a crucial skill in these and many other areas.

Further Exploration: The Discriminant

The expression inside the square root in the quadratic formula (b² - 4ac) is called the discriminant. It provides valuable information about the nature of the solutions:

• b² - 4ac > 0: Two distinct real solutions (as we saw in our example).
• b² - 4ac = 0: One real solution (a repeated root).
• b² - 4ac < 0: Two complex conjugate solutions (involving imaginary numbers).

Analyzing the discriminant before attempting to solve the equation can help predict the nature of the solutions and guide the choice of solution method. In our example, b² - 4ac = 288 > 0, indicating two distinct real solutions.

Conclusion

Solving the quadratic equation x² + 12x - 36 = 0 demonstrates the power and versatility of different algebraic techniques. While factoring provided simple integer solutions in this specific case, completing the square and the quadratic formula offer more general approaches that handle a wider range of quadratic equations, including those with irrational or complex solutions. Understanding these methods and their underlying principles is essential for anyone working with quadratic equations in various mathematical and real-world applications. The choice of method often depends on personal preference and the nature of the specific equation encountered. However, a solid grasp of all three methods ensures a well-rounded understanding of this fundamental mathematical concept.