Solution of Linear and Quadratic Equations

Solution of Linear and Quadratic Equations

Linear Equations

  • A linear equation is an equation where the highest degree (or power) of the variable (commonly x) is one.
  • Linear equations have the form ax + b = 0; where a is the coefficient of x, b is the constant and a ≠ 0.
  • Solutions of linear equations are found by isolating x on one side of the equation.

Quadratic Equations

  • A quadratic equation is an equation in which the highest degree (or power) of the variable (commonly x) is 2.
  • Quadratic equations have the form ax^2 + bx + c = 0; where a is the coefficient of x^2, b is the coefficient of x, c is the constant and a ≠ 0.
  • There are three methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.

Factoring

  • Factoring to solve for a quadratic equation is the process of writing the equation as two binomial expressions set equal to zero. Once factored, you can set each factor equal to zero and solve for x.
  • Factoring is best used when all terms are divisible by a common factor or when the equation can be written as a perfect square trinomial.

Quadratic Formula Method

  • The Quadratic Formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a). This method is beneficial for all types of quadratic equations, especially when factoring is not favorable.

Completing the Square

  • Completing the square method involves rewriting the quadratic equation so that the left-hand side is a perfect square trinomial. This will allow you to write the equation in the form (x - p)^2 = q.
  • This method allows us to derive the quadratic formula and is useful for understanding the graphical transformation of quadratic functions.

Remember: All three methods should lead to the same solutions for x. It is critical to familiarize yourself with each method so that you can select the most efficient method based on the given quadratic equation.