Determine The Solution For X2 36 12x

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

This article will guide you through various methods of solving the quadratic equation x² + 12x - 36 = 0. Understanding how to solve quadratic equations is fundamental in algebra and has wide applications in various fields. On top of that, we'll explore different approaches, from factoring and using the quadratic formula to completing the square, explaining each step clearly and providing examples to enhance your understanding. This complete walkthrough will equip you with the skills to confidently tackle similar problems.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. It generally takes the form 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.

Solving a quadratic equation means finding the values of x that make the equation true. On top of that, these values are called the roots or solutions of the equation. Quadratic equations can have two, one, or no real solutions, depending on the discriminant (b² - 4ac). We'll explore this further later.

Method 1: Factoring

Factoring is a method of solving quadratic equations by rewriting the equation as a product of two simpler expressions. If we can find two numbers that add up to b (12 in our case) and multiply to c (-36), we can factor the equation.

Let's find those numbers. We need two numbers that add to 12 and multiply to -36. After some trial and error (or using a systematic approach like listing factors of -36), we find that 18 and -6 satisfy these conditions: 18 + (-6) = 12 and 18 * (-6) = -36.

So, we can rewrite the equation as:

(x + 18)(x - 6) = 0

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

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

That's why, the solutions to the equation x² + 12x - 36 = 0 are x = -18 and x = 6.

Method 2: Quadratic Formula

The quadratic formula provides a direct way to solve any quadratic equation, regardless of whether it's easily factorable. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 12, c = -36):

x = [-12 ± √(12² - 4 * 1 * -36)] / (2 * 1) x = [-12 ± √(144 + 144)] / 2 x = [-12 ± √288] / 2 x = [-12 ± √(144 * 2)] / 2 x = [-12 ± 12√2] / 2 x = -6 ± 6√2

This gives us two solutions:

• x = -6 + 6√2
• x = -6 - 6√2

These solutions are equivalent to the solutions obtained through factoring. The difference lies in the form of the solution; the quadratic formula provides solutions in their exact form, which might involve irrational numbers.

Method 3: Completing the Square

Completing the square is another powerful method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

1. Move the constant term to the right side:

   x² + 12x = 36

2. Take half of the coefficient of x (which is 12), square it (6² = 36), and add it to both sides:

   x² + 12x + 36 = 36 + 36 x² + 12x + 36 = 72

3. Factor the left side as a perfect square:

   (x + 6)² = 72

4. Take the square root of both sides:

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   x + 6 = ±√72 x + 6 = ±√(36 * 2) x + 6 = ±6√2

5. Solve for x:

   x = -6 ± 6√2

This yields the same solutions as the quadratic formula: x = -6 + 6√2 and x = -6 - 6√2.

Understanding the Discriminant (b² - 4ac)

The discriminant, b² - 4ac, provides valuable information about the nature of the solutions of a quadratic equation:

• If b² - 4ac > 0: The equation has two distinct real roots (as in our case).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has no real roots; the roots are complex numbers (involving the imaginary unit i).

In our equation, b² - 4ac = 12² - 4(1)(-36) = 144 + 144 = 288 > 0, indicating two distinct real roots.

Graphical Representation

The solutions of a quadratic equation represent the x-intercepts (where the graph intersects the x-axis) of the parabola represented by the equation y = x² + 12x - 36. The parabola opens upwards since the coefficient of x² (a = 1) is positive. The x-intercepts are at x = -18 and x = 6.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains.
• Economics: Modeling supply and demand, calculating optimal production levels.
• Computer graphics: Representing curves and shapes.

Frequently Asked Questions (FAQ)

• Q: Can I always factor a quadratic equation? A: No, not all quadratic equations can be easily factored using integer coefficients. The quadratic formula or completing the square are more general methods.

• Q: What if the coefficient of x² (a) is not 1? A: You can still use the quadratic formula or completing the square. Factoring might become more challenging, but it's still a possible approach.

• Q: What do the solutions represent in a real-world context? A: The solutions depend on the context of the problem. They could represent time, distance, quantity, or other relevant variables.

• Q: Why are there sometimes two solutions? A: A parabola (the graph of a quadratic equation) can intersect the x-axis at two points, representing two distinct values of x that satisfy the equation.

• Q: What does it mean when there are no real solutions? A: This means the parabola does not intersect the x-axis. The solutions are complex numbers, involving the imaginary unit 'i'.

Conclusion

Solving quadratic equations is a fundamental skill in algebra. We've explored three main methods: factoring, the quadratic formula, and completing the square. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and your preference. Understanding the discriminant helps predict the nature of the solutions. Now, remember, practice is key to mastering these techniques. Think about it: by working through various examples and applying these methods, you'll develop a strong understanding of quadratic equations and their applications. This knowledge will serve as a solid foundation for tackling more complex mathematical problems in the future.