x 2 3x 2 0

Guri Bee

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21-10-2024

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Unpacking the Quadratic: x² + 2x + 3x² = 0

This article explores the equation x² + 2x + 3x² = 0, a simple example of a quadratic equation. We'll break down its components, solve for x, and provide a visual representation.

Understanding the Equation

• Quadratic Equation: This equation is classified as a quadratic because its highest power of the variable 'x' is 2 (x²).
• Standard Form: Before we solve, let's rearrange the terms to get it into the standard quadratic form (ax² + bx + c = 0):
  • 4x² + 2x = 0
• Coefficients: We can identify the coefficients:
  • a = 4
  • b = 2
  • c = 0 (since there is no constant term)

Solving for x

There are multiple ways to solve quadratic equations:

1. Factoring:

   • Step 1: Find the greatest common factor (GCF) of the terms. Here, the GCF is 2x.
   • Step 2: Factor out the GCF: 2x(2x + 1) = 0
   • Step 3: Set each factor to zero and solve for x:
     • 2x = 0 --> x = 0
     • 2x + 1 = 0 --> x = -1/2

2. Quadratic Formula: This formula is a general solution for any quadratic equation.

   • Step 1: Identify the values of a, b, and c (from the standard form).
   • Step 2: Substitute the values into the formula:
     • x = (-b ± √(b² - 4ac)) / 2a
   • Step 3: Simplify and solve for x.

Visual Representation

The graph of this equation would be a parabola, crossing the x-axis at the points x = 0 and x = -1/2. The parabola opens upwards because the coefficient of the x² term (a = 4) is positive.

Further Exploration:

• Discriminant: The expression inside the square root of the quadratic formula (b² - 4ac) is called the discriminant. It tells us about the nature of the solutions:
  • If the discriminant is positive, there are two distinct real solutions.
  • If the discriminant is zero, there is one repeated real solution.
  • If the discriminant is negative, there are two complex solutions.
• Applications: Quadratic equations are essential for solving various real-world problems in fields like physics, engineering, and finance.

Conclusion:

The equation x² + 2x + 3x² = 0 is a simple example of a quadratic equation, which can be solved through factoring or using the quadratic formula. Understanding these methods allows us to solve more complex equations and apply them to real-world scenarios.