Completing the Square Notes

Factoring Trinomials and Completing the Square

Factoring Basics

  • Factoring involves finding two numbers that multiply to c and add up to b.
  • If a trinomial cannot be factored, it implies no such integer values exist that satisfy both conditions simultaneously.

Steps for Completing the Square

  1. Move the c term: Isolate the x^2 and x terms on one side of the equation.
  2. Complete the square: Divide the b term by 2, square it, and add it to both sides of the equation. This fills in the boxes, creating a perfect square trinomial. Mathematically, this involves calculating \left(\frac{b}{2}\right)^2 and adding it to both sides.
  3. Factor the trinomial: Factor the perfect square trinomial into a binomial squared, (x + a)^2 where a = \frac{b}{2}.

Avoiding Common Mistakes

  • It's crucial to maintain the balance of the equation. Any operation performed on one side must also be done on the other.
  • The goal is to manipulate the equation into a form where isolating x becomes straightforward.
  • If the equation isn't set to zero, you can't split the terms.

Solving for x

  1. Take the square root of both sides to undo the square, remembering to include both positive and negative roots.
    • For example: If (x - 3)^2 = 21, then x - 3 = \pm \sqrt{21}.
  2. Isolate x to find the solutions.
    • So, x = 3 \pm \sqrt{21}.

Nature of Solutions

  • The solutions obtained (3 + \sqrt{21} and 3 - \sqrt{21}) are irrational numbers because the square root of 21 is irrational.
  • Solutions can be expressed separately or combined using the plus-minus symbol.

Simplifying Radicals

  • Simplifying radicals involves finding perfect square factors within the radicand (the value inside the square root).
  • If asked to provide the answer in simplest radical form, identify and extract any perfect square factors.
  • For example, if you have a radical like \sqrt{52}, look for perfect square factors of 52.

Example

  • Starting with an expression like x^2 + bx + c = 0 perform the following
    • Move the c term: x^2 + bx = -c
    • Calculate \left(\frac{b}{2}\right)^2 and add to both sides.
    • The result of the left side of the equation yields a perfect square trinomial.
    • Factor the perfect square trinomial into (x + a)^2 where a = \frac{b}{2}.