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 $a = \frac{b}{2}$.