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Linear and quadratic (Posted on 2007-10-23) Difficulty: 2 of 5
Solve bx + c = 0 for x by means of the quadratic formula.

See The Solution Submitted by Kurious    
Rating: 3.0000 (3 votes)

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My 3 solutions. | Comment 8 of 9 |
Haven't looked at the others but these 3 occurred to me.  Non of them are really calculus:

1)  Multiply both sides of the equation by x.
Results in bx^2 + cx = 0 which can be solved by the quadratic formula.  x = (-c +/- c)/2b
Solution x=0 can be discarded leaving x=-c/b

2)  Square both sides of the equation.
Results in b^2x^2 + 2bcx + c^2 = 0 which can be solved by the quad. form.  x = (-2bc +/- 0)/(2b^2)
Solution is a double root x=-c/b

3) Substitute y^2 for x.
Results in by^2 + c  which can be solved by the quad. form.
y = (0 +/- sqrt(-4bc))/2b = +/- sqrt(-bc)/b = +/- i sqrt (bc)/b
y^2 = -b^2c^2/b^2 = -c/b

(I realize the first two can be generalized to multiplying both sides by x - h and discarding the extraneous root x = h)

  Posted by Jer on 2007-10-24 08:31:53
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