Completing the Square Notes

Completing the Square to Solve Quadratic Functions

  • Definition: Completing the square is a method to convert a quadratic expression into a perfect square trinomial.

Steps to Complete the Square:

  1. Find the term needed to complete the square, which is calculated as (b/2)², where b is the coefficient of x in the expression x² + bx.
  2. Add this term to the quadratic expression.

Example:

  • Find the value of c to make x² + 4x + c a perfect square trinomial:
  • (4/2)² = 2² = 4
  • Thus, c = 4, making the expression: x² + 4x + 4 = (x + 2)².

Additional Examples:

  • For x² + 8x + c:
  • (8/2)² = 16; so c = 16 → (x + 4)².
  • For x² + 9x + c:
  • (9/2)² = 20.25; so c = 20.25 → (x + 4.5)².

Solving Quadratic Equations by Completing the Square:

  1. Isolate the x² + bx term on one side of the equation.
  2. Find the term (b/2)² and add to both sides of the equation.
  3. Factor the left side and simplify the right side.
  4. Take the square root of both sides and solve for x.

Example 1:

  • Solve x² - 6x + 12 = 0:
  • Isolate: x² - 6x = -12
  • Complete the square: (6/2)² = 9 → x² - 6x + 9 = -12 + 9
  • This leads to (x - 3)² = -3 (no real solution).

Example 2:

  • Solve 3x² - 9x = 21:
  • Rearrange: 3x² - 9x - 21 = 0 → x² - 3x = 7
  • Complete the square for x² - 3x: (3/2)² = 2.25
  • Add to both sides: x² - 3x + 2.25 = 7 + 2.25
  • Factor: (x - 1.5)² = 9.25
  • Solve for x: x - 1.5 = ±√9.25.

Practical Application Example:

  • Football Jerseys Cost Model: C = 0.1x² + 2.4x + 25, where C is the cost and x is the number of jerseys.
  • To find how many jerseys can be purchased with $430:
  • Set up the equation: 430 = 0.1x² + 2.4x + 25
  • Rearrange to form a standard quadratic equation: 0.1x² + 2.4x - 405 = 0
  • Use completing the square to find x, leading to a solution of approximately 52.8 or 52 jerseys.

More Examples of Completing the Square:

  • Expression: x² - 4x + c → determine c that makes (x - 2)² = 4 + c.
  • Expression: x² + 15x + c → (15/2)² = 56.25; thus, c = 56.25.

Quadratic Equations:

  • To solve x² - 4x = 21:
  • Rearranged to x² - 4x - 21 = 0.
  • Then apply completing the square method to find x.

Note on Real Solutions:

  • Not all quadratics yield real solutions; if the discriminant (b² - 4ac) is negative, the equation has no real solutions.

Summary of Key Steps:

  1. Identify the quadratic and isolate.
  2. Calculate (b/2)² and complete the square.
  3. Solve for x using the square root method.
  4. Interpret the solutions based on the context of the problem.