What is what does it mean to complete the square?

Completing the square is a technique used in algebra to convert a quadratic expression into a perfect square trinomial, plus a constant term. This process is particularly useful for solving quadratic equations, graphing parabolas, and simplifying expressions.

Here's a breakdown of what it entails:

• What is a Perfect Square Trinomial? A perfect square trinomial is a quadratic expression that can be factored into the form (ax + b)² or (ax - b)².

• The General Form: Starting with a quadratic expression in the form of ax² + bx + c, the goal of completing the square is to rewrite it as a(x + h)² + k, where h and k are constants.

• The Steps:

  1. Ensure the Leading Coefficient is 1: If a ≠ 1, factor out 'a' from the x² and x terms.
  2. Find the Value to Add and Subtract: Take half of the coefficient of the x term (b/2), square it ((b/2)²), and add and subtract this value inside the expression (or inside the parentheses if you factored out 'a'). This maintains the expression's original value.
  3. Form the Perfect Square Trinomial: The first three terms (x² + bx + (b/2)²) will form a perfect square trinomial that can be factored into (x + b/2)².
  4. Simplify: Simplify the expression by combining the remaining constant terms. Remember to distribute the 'a' if you factored it out in the first step.

• Why is it Useful?

  • Solving Quadratic Equations: Completing the square allows you to solve quadratic equations by isolating the squared term and then taking the square root of both sides. This is an alternative to using the quadratic formula.
  • Finding the Vertex of a Parabola: When a quadratic equation is in the form a(x + h)² + k, the vertex of the corresponding parabola is at the point (-h, k). This is very important for graphing parabolas. Learn more about parabolas.
  • Integration: Completing the square can be helpful in simplifying integrals, especially when dealing with rational functions.

• Example:

  Let's complete the square for x² + 6x + 5.

  1. Coefficient of x² is already 1.
  2. Half of 6 is 3, and 3² is 9. Add and subtract 9: x² + 6x + 9 - 9 + 5
  3. Factor the perfect square trinomial: (x + 3)² - 9 + 5
  4. Simplify: (x + 3)² - 4

  So, x² + 6x + 5 = (x + 3)² - 4. The vertex of the parabola represented by this equation is at (-3, -4).