Understanding First and Second-Degree Equations

Introduction

Equations are fundamental in mathematics, serving as the basis for solving various problems. In this lesson, we will explore first-degree and second-degree equations, understanding their structure and how to solve them.

First-Degree Equations

A first-degree equation, also known as a linear equation, is an equation of the form ax + b = 0, where a and b are constants, and x is the variable. The highest power of the variable is one, which makes it a linear equation.

Example

Consider the equation 2x + 3 = 0. To solve for x, follow these steps:

  1. Subtract 3 from both sides: 2x = -3
  2. Divide both sides by 2: x = -3/2

Thus, the solution is x = -1.5.

Second-Degree Equations

A second-degree equation, or quadratic equation, is of the form ax2 + bx + c = 0, where a, b, and c are constants, and x is the variable. The highest power of the variable is two.

Example

Consider the equation x2 - 5x + 6 = 0. To solve this, we can use the factorization method:

  1. Factor the quadratic: (x - 2)(x - 3) = 0
  2. Set each factor to zero: x - 2 = 0 or x - 3 = 0
  3. Solve for x: x = 2 or x = 3

Thus, the solutions are x = 2 and x = 3.

Solving Quadratic Equations

There are several methods to solve quadratic equations:

  • Factoring: As shown in the example, factor the equation and solve for x.
  • Quadratic Formula: Use the formula x = (-b ± √(b2 - 4ac)) / 2a to find the solutions.
  • Completing the Square: Rearrange the equation to form a perfect square trinomial.

Conclusion

Understanding and solving first-degree and second-degree equations are crucial skills in mathematics. These equations form the foundation for more complex mathematical concepts and real-world problem-solving. Practice solving different types of equations to become proficient in these essential skills.

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