5x 2 11x 2 0

Decoding the Mystery: 5x² + 11x + 2 = 0 A Comprehensive Guide to Quadratic Equations

This article will delve into the solution and deeper meaning behind the quadratic equation 5x² + 11x + 2 = 0. We will explore various methods to solve this equation, explain the underlying mathematical principles, and provide a thorough understanding of quadratic equations in general. This includes understanding the concept of roots, the discriminant, and the different approaches to finding solutions, making this a complete guide suitable for students and anyone interested in strengthening their mathematical skills.

Introduction: Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 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 would be a linear equation). Our specific equation, 5x² + 11x + 2 = 0, fits this form with a = 5, b = 11, and c = 2. Understanding how to solve this type of equation is fundamental to algebra and has numerous applications in various fields like physics, engineering, and computer science.

Method 1: Factoring

Factoring is a technique to rewrite the quadratic expression as a product of two simpler expressions. This method is particularly useful when the equation can be easily factored. Let's try factoring our equation:

5x² + 11x + 2 = 0

We are looking for two numbers that add up to 11 (the coefficient of x) and multiply to 10 (the product of a and c, which is 5 * 2 = 10). These numbers are 10 and 1. We can rewrite the equation as:

5x² + 10x + x + 2 = 0

Now we can factor by grouping:

5x(x + 2) + 1(x + 2) = 0

(5x + 1)(x + 2) = 0

This gives us two possible solutions:

5x + 1 = 0 or x + 2 = 0

Solving for x:

x = -1/5 or x = -2

Therefore, the solutions to the equation 5x² + 11x + 2 = 0 are x = -1/5 and x = -2. These are also known as the roots or zeros of the quadratic equation.

Method 2: Quadratic Formula

The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it's easily factorable or not. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 5, b = 11, c = 2):

x = [-11 ± √(11² - 4 * 5 * 2)] / (2 * 5)

x = [-11 ± √(121 - 40)] / 10

x = [-11 ± √81] / 10

x = [-11 ± 9] / 10

This gives us two solutions:

x = (-11 + 9) / 10 = -2/10 = -1/5

x = (-11 - 9) / 10 = -20/10 = -2

Again, we arrive at the same solutions: x = -1/5 and x = -2.

Method 3: Completing the Square

Completing the square is another algebraic method to solve quadratic equations. This method involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. While less efficient than the quadratic formula for this particular problem, understanding this method provides a deeper insight into the structure of quadratic equations.

Let's complete the square for 5x² + 11x + 2 = 0:

1. Divide the equation by a (5): x² + (11/5)x + 2/5 = 0

2. Move the constant term to the right side: x² + (11/5)x = -2/5

3. Take half of the coefficient of x (11/5), square it ((11/10)² = 121/100), and add it to both sides: x² + (11/5)x + 121/100 = -2/5 + 121/100

4. Factor the left side as a perfect square: (x + 11/10)² = (-40 + 121)/100 = 81/100

5. Take the square root of both sides: x + 11/10 = ±√(81/100) = ±9/10

6. Solve for x: x = -11/10 ± 9/10

This again yields the solutions x = -1/5 and x = -2.

Understanding the Discriminant (b² - 4ac)

The expression b² - 4ac within the quadratic formula is called the discriminant. It provides valuable information about the nature of the roots of the quadratic equation:

• If b² - 4ac > 0: The equation has two distinct real roots (as in our example).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex roots (roots involving imaginary numbers).

In our case, b² - 4ac = 121 - 40 = 81, which is positive, confirming that we have two distinct real roots.

Graphical Representation

The solutions to the quadratic equation 5x² + 11x + 2 = 0 represent the x-intercepts of the parabola represented by the function y = 5x² + 11x + 2. Graphically, this means the parabola intersects the x-axis at x = -1/5 and x = -2. The parabola opens upwards because the coefficient of x² (a = 5) is positive.

Applications of Quadratic Equations

Quadratic equations have widespread applications in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stress and strain, solving circuit problems.
• Economics: Modeling supply and demand curves, optimizing production.
• Computer Science: Algorithm design, computer graphics, optimization problems.

Frequently Asked Questions (FAQ)

• Q: What if the quadratic equation cannot be easily factored?

  • A: Use the quadratic formula. It works for all quadratic equations, regardless of their factorability.

• Q: What are complex roots?

  • A: Complex roots involve imaginary numbers (represented by i, where i² = -1). They occur when the discriminant (b² - 4ac) is negative.

• Q: Can a quadratic equation have only one solution?

  • A: Yes, this happens when the discriminant is equal to zero. The single solution is often referred to as a repeated root.

• Q: How can I check if my solutions are correct?

  • A: Substitute the solutions back into the original equation. If the equation holds true, your solutions are correct. For example, substituting x = -1/5 into 5x² + 11x + 2 = 0 should result in 0 = 0.

• Q: Why is 'a' not allowed to be zero in a quadratic equation?

  • A: If a were zero, the x² term would vanish, and the equation would become a linear equation (bx + c = 0), which has a different solution method.

Conclusion: Mastering Quadratic Equations

Solving the quadratic equation 5x² + 11x + 2 = 0, whether through factoring, the quadratic formula, or completing the square, demonstrates the fundamental principles of solving second-degree polynomial equations. Understanding these methods and the concept of the discriminant is crucial for tackling more advanced mathematical problems. The application of quadratic equations extends far beyond the realm of theoretical mathematics, playing a significant role in numerous scientific and engineering disciplines. This comprehensive guide aims to equip you with the knowledge and tools to confidently solve any quadratic equation you encounter. Remember to practice regularly to solidify your understanding and build your problem-solving skills.