What three techniques can be used to solve a quadratic equation?Demonstrate these techniques on the equation "12x2 - 10x - 42 = 0."

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Oddman answered
1. Factor it.
  12x^2 - 10x - 42 = 0
  12x^2 - 28x + 18x - 42 = 0
  4x(3x - 7) + 6(3x - 7) = 0
  (4x + 6)(3x - 7) = 0
  x = {-6/4, 7/3} = {-3/2, 7/3}

2. Use the quadratic equation.
  X = (-b ±√(b^2-4ac))/(2a)
  = (-(-10) ±√((-10)^2 - 4(12)(-42)))/(2(12))    (substitute a=12, b=-10, c=-42)
  = (10 ±√(100 + 2016))/24    (evaluate)
  = (5 ± 23)/12    (evaluate, reduce)
  x = {-3/2, 7/3}    (evaluate, reduce)

3. Complete the square. (Equivalent to using the quadratic equation.)
  12x^2 - 10x - 42 = 0
  x^2 - 5/6x = 7/2    (add 42, divide by 12, reduce)
  x^2 - 5/6x + 25/144 = 7/2 + 25/144    (add the square of half of 5/6)
  (x - 5/12)^2 = 529/144    (show the completed square, evaluate right side)
  x - 5/12 = ±23/12    (square root)
  x = (5 ± 23)/12    (add 5/12, same answer as above)

Other techniques can also be used to solve a quadratic, including
  - graph it
  - use an iteration technique, such as Newton's iteration   - use trial and error, with the Rational Root Theorem roots as starting points.

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