X 2 2x 80 0

Decoding the Mystery: Understanding the Equation x² + 2x = 80

This article delves into the solution of the quadratic equation x² + 2x = 80, exploring various methods to solve it and explaining the underlying mathematical principles. We'll move beyond simply finding the solution to understand the nature of quadratic equations, their applications, and how to approach similar problems effectively. This comprehensive guide is designed for students and anyone seeking a deeper understanding of algebra. We will cover multiple approaches, ensuring a robust and thorough understanding of the problem.

Introduction: What is a Quadratic Equation?

Before we tackle the specific equation x² + 2x = 80, let's establish a foundational understanding. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it wouldn't be a quadratic equation). Our equation, x² + 2x = 80, needs to be rearranged into this standard form before we can effectively solve it.

Rearranging the Equation into Standard Form

The first step in solving x² + 2x = 80 is to rearrange it into the standard quadratic equation format (ax² + bx + c = 0). To do this, we subtract 80 from both sides of the equation:

x² + 2x - 80 = 0

Now we have the equation in the standard form, where a = 1, b = 2, and c = -80. This form allows us to apply various solution methods.

Method 1: Factoring

Factoring is a method of solving quadratic equations by expressing the quadratic expression as a product of two linear expressions. We are looking for two numbers that add up to 'b' (2) and multiply to 'c' (-80). These numbers are 10 and -8. Therefore, we can factor the equation as follows:

(x + 10)(x - 8) = 0

This equation is true if either (x + 10) = 0 or (x - 8) = 0. Solving these linear equations gives us the two solutions:

x + 10 = 0 => x = -10 x - 8 = 0 => x = 8

Therefore, the solutions to the equation x² + 2x = 80 are x = -10 and x = 8.

Method 2: Quadratic Formula

The quadratic formula is a more general method that works for all quadratic equations, even those that are difficult or impossible to factor. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 2, c = -80), we get:

x = [-2 ± √(2² - 4 * 1 * -80)] / (2 * 1) x = [-2 ± √(4 + 320)] / 2 x = [-2 ± √324] / 2 x = [-2 ± 18] / 2

This gives us two solutions:

x = (-2 + 18) / 2 = 16 / 2 = 8 x = (-2 - 18) / 2 = -20 / 2 = -10

Again, we obtain the solutions x = 8 and x = -10. The quadratic formula provides a reliable method to solve any quadratic equation.

Method 3: Completing the Square

Completing the square is another algebraic technique to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. Let's apply this method to our equation:

x² + 2x - 80 = 0

1. Move the constant term to the right side: x² + 2x = 80

2. Take half of the coefficient of x (which is 2), square it (1), and add it to both sides: x² + 2x + 1 = 80 + 1 x² + 2x + 1 = 81

3. The left side is now a perfect square trinomial: (x + 1)² = 81

4. Take the square root of both sides: x + 1 = ±√81 x + 1 = ±9

5. Solve for x: x = -1 + 9 = 8 x = -1 - 9 = -10

Once again, we arrive at the solutions x = 8 and x = -10.

Graphical Representation

The solutions to the quadratic equation represent the x-intercepts (where the graph intersects the x-axis) of the parabola represented by the equation y = x² + 2x - 80. Graphing the equation visually confirms our solutions. The parabola will cross the x-axis at x = 8 and x = -10.

The Discriminant and Nature of Roots

The expression inside the square root in the quadratic formula (b² - 4ac) is called the discriminant. It determines the nature of the roots (solutions) of the quadratic equation:

• b² - 4ac > 0: Two distinct real roots (as we have in this case).
• b² - 4ac = 0: One real root (a repeated root).
• b² - 4ac < 0: Two complex conjugate roots (no real solutions).

In our equation, the discriminant is 2² - 4(1)(-80) = 324, which is greater than 0, indicating two distinct real roots, which are 8 and -10.

Applications of Quadratic Equations

Quadratic equations have widespread applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing bridges, buildings, and other structures.
• Economics: Modeling supply and demand, analyzing market trends.
• Computer Graphics: Creating curves and shapes.

Frequently Asked Questions (FAQ)

Q: Can I use a calculator to solve this equation?

A: Yes, many scientific calculators have built-in functions to solve quadratic equations. You can input the values of a, b, and c and the calculator will provide the solutions. However, understanding the underlying methods is crucial for a deeper grasp of the mathematics involved.

Q: What if the equation is more complex?

A: For more complex quadratic equations, the quadratic formula is the most reliable method. Factoring might not always be straightforward, and completing the square can become cumbersome.

Q: What are complex roots?

A: Complex roots involve imaginary numbers (represented by 'i', where i² = -1). They arise when the discriminant (b² - 4ac) is negative. These roots are not directly representable on a standard Cartesian plane.

Q: Why are there always two solutions (or sometimes one repeated solution)?

A: A quadratic equation represents a parabola, and a parabola can intersect the x-axis at most at two points. These points of intersection represent the solutions to the equation. A single repeated solution occurs when the parabola just touches the x-axis at one point.

Conclusion: Mastering Quadratic Equations

Solving x² + 2x = 80, while seemingly simple, provides a gateway to understanding the broader world of quadratic equations. Through factoring, the quadratic formula, and completing the square, we've demonstrated multiple paths to arrive at the same solutions: x = 8 and x = -10. Mastering these methods is crucial not only for academic success but also for applying this essential mathematical concept to various real-world scenarios. Remember to always focus on understanding the underlying principles to successfully tackle even more complex algebraic challenges. The ability to manipulate equations, interpret results, and understand the significance of the discriminant are vital skills for any aspiring mathematician or scientist.