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Seven Weeks to Precalculus: Day 1 – Quadratic Equations

May 7th, 2012 | By | Category: Featured Article, snewz

General form of quadratic equations:

a, b, and c are all elements of the set of real numbers (a, b, & c ∈ of ℝ)

a ≠ 0

ax2: Quadratic Term

bx: Linear Term

c: Constant Term

Three fundamental truths of quadratic equations:

  1. Quadratic equations are also called polynomial equations (exponential equations whose exponents are raised to integers (positive whole #’s)).
  2. The solutions are the values that make the equation true
  3. There are two solutions to a quadratic equation
    1. Two real solutions (ex: x2-9=0; x=±3)
    2. Two complex solutions (ex: x2+9=0; x=±3i)
    3. One real solution with multiplicity 2 (ex: x2-2x+1=0=(x-1)2; x=1±0)

Methods for solving Quadratic Equations:

  1. Factor the quadratic equation into linear factors (zero product property)

“If a times b = 0, this implies a=0, b=0, or a&b = 0.”

  1. Square Root Property: Rearrange terms so that your variable squared is equal to a number.
    1. x2=k
    2. x=±√k
    3. Completing the square: Rearrange terms and add a value to both sides of the equation so that one factor with multiplicity 2 equals a real number.

  1. Quadratic Formula: Derived from ax^2+bx+c format. See khanacademy.org link for step-by-step proof: Khan Academy Quadratic Formula Proof

 

 


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