Solving Quadratic Equations Using The Quadratic Formula

by ADMIN 56 views

Introduction

The quadratic formula is a fundamental tool in algebra used to solve quadratic equations, which are equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations arise in various fields, including physics, engineering, economics, and computer science. The quadratic formula provides a universal method for finding the solutions (also called roots or zeros) of any quadratic equation, regardless of whether it can be easily factored or not. In this comprehensive guide, we will delve into the intricacies of the quadratic formula, explore its derivation, and demonstrate its application through step-by-step examples. We will also discuss the discriminant, a crucial component of the quadratic formula that reveals the nature of the solutions.

Understanding Quadratic Equations

Before diving into the quadratic formula itself, it's essential to grasp the concept of 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 is the coefficient of the x² term (the quadratic coefficient).
  • b is the coefficient of the x term (the linear coefficient).
  • c is the constant term.

The solutions to a quadratic equation are the values of x that satisfy the equation, i.e., the values that make the equation true. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis. A quadratic equation can have two distinct real solutions, one real solution (a repeated root), or two complex solutions.

The Quadratic Formula: A Powerful Tool

The quadratic formula is a mathematical formula that provides the solutions to any quadratic equation in the standard form ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

Where:

  • x represents the solutions (roots) of the quadratic equation.
  • a, b, and c are the coefficients from the quadratic equation.
  • The symbol ± indicates that there are two possible solutions: one with addition (+) and one with subtraction (-).
  • The expression inside the square root, b² - 4ac, is called the discriminant, which plays a crucial role in determining the nature of the solutions.

Derivation of the Quadratic Formula

The quadratic formula is derived using a technique called completing the square. This method involves manipulating the quadratic equation to create a perfect square trinomial on one side of the equation. Here's a step-by-step outline of the derivation:

  1. Start with the standard form of the quadratic equation: ax² + bx + c = 0
  2. Divide both sides by a (assuming a ≠ 0): x² + (b/ a)x + (c/ a) = 0
  3. Move the constant term (c/ a) to the right side: x² + (b/ a)x = -(c/ a)
  4. Complete the square on the left side. To do this, take half of the coefficient of the x term (b/ a), square it, and add it to both sides. Half of (b/ a) is (b/ 2a), and squaring it gives (b² / 4a²): x² + (b/ a)x + (b² / 4a²) = -(c/ a) + (b² / 4a²)
  5. Rewrite the left side as a perfect square and simplify the right side: (x + (b/ 2a))² = (b² - 4ac) / (4a²)
  6. Take the square root of both sides: x + (b/ 2a) = ±√((b² - 4ac) / (4a²))
  7. Simplify the square root: x + (b/ 2a) = ±√( b² - 4ac ) / (2a)
  8. Isolate x by subtracting (b/ 2a) from both sides: x = -(b/ 2a) ± √( b² - 4ac ) / (2a)
  9. Combine the terms on the right side to get the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

Applying the Quadratic Formula: A Step-by-Step Guide

To use the quadratic formula effectively, follow these steps:

  1. Write the quadratic equation in standard form: Ensure the equation is in the form ax² + bx + c = 0. If necessary, rearrange the terms to achieve this form.

  2. Identify the coefficients a, b, and c: Carefully determine the values of a, b, and c from the equation.

  3. Substitute the values into the quadratic formula: Plug the values of a, b, and c into the formula: x = (-b ± √(b² - 4ac)) / (2a)

  4. Simplify the expression: Evaluate the expression under the square root (the discriminant) and simplify the entire formula as much as possible.

  5. Calculate the two solutions: Compute the two possible values of x, one using the plus sign (+) and one using the minus sign (-) in front of the square root.

  6. Express the solutions: Write the solutions clearly, either as separate values or as a comma-separated list.

Example Problem: Solving 2x(x + 3) = -1

Let's apply the quadratic formula to solve the equation 2x(x + 3) = -1. This example will walk you through each step of the process, providing a clear understanding of how to use the formula effectively.

  1. Write the equation in standard form: First, we need to expand and rearrange the equation to get it into the form ax² + bx + c = 0. 2x(x + 3) = -1 2x² + 6x = -1 2x² + 6x + 1 = 0

  2. Identify the coefficients a, b, and c: From the standard form, we can identify the coefficients:

    • a = 2
    • b = 6
    • c = 1

  3. Substitute the values into the quadratic formula: Now, we substitute these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) x = (-6 ± √(6² - 4 * 2 * 1)) / (2 * 2)

  4. Simplify the expression: Next, we simplify the expression step by step: x = (-6 ± √(36 - 8)) / 4 x = (-6 ± √28) / 4 x = (-6 ± 2√7) / 4

  5. Reduce the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2: x = (-3 ± √7) / 2

  6. Calculate the two solutions: We now have two possible solutions:

    • x₁ = (-3 + √7) / 2
    • x₂ = (-3 - √7) / 2

  7. Express the solutions: The solutions to the equation 2x(x + 3) = -1 are: x = (-3 + √7) / 2, (-3 - √7) / 2

Therefore, the solutions to the quadratic equation 2x(x + 3) = -1, expressed as a comma-separated list, are ((-3 + √7) / 2, (-3 - √7) / 2).

The Discriminant: Unveiling the Nature of Solutions

The discriminant, denoted as Δ, is the expression inside the square root in the quadratic formula: Δ = b² - 4ac. The discriminant provides valuable information about the nature of the solutions to the quadratic equation.

  1. Δ > 0 (Positive Discriminant): If the discriminant is positive, the quadratic equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.

  2. Δ = 0 (Zero Discriminant): If the discriminant is zero, the quadratic equation has one real solution (a repeated root). This means the parabola touches the x-axis at exactly one point (the vertex).

  3. Δ < 0 (Negative Discriminant): If the discriminant is negative, the quadratic equation has two complex solutions (also called imaginary solutions). This means the parabola does not intersect the x-axis.

Conclusion

The quadratic formula is an indispensable tool for solving quadratic equations. Its ability to provide solutions for any quadratic equation, regardless of its factorability, makes it a cornerstone of algebra. By understanding the formula, its derivation, and the significance of the discriminant, you can confidently tackle a wide range of quadratic equation problems. Remember to follow the steps carefully, paying close attention to the signs and coefficients, and you'll master the art of solving quadratic equations using the quadratic formula.