The Goal
Solve the following equation for c:
The Mistake
Find the algebra mistake:
(Roll the mouse over the algebra to see a hint in red)
The Correction
(Roll the mouse over the area above to see the correction in blue)
An Explanation
(In this explanation we cannot use c for the constant coefficient of a general quadratic since the variable in the quadratic is c. We use d instead.)
It is no help in solving a quadratic equation to arrange it in the form ac^{2} + bc = d (unless d = 0) (in this problem, c^{2} + 7c = 6) and then write it in the form c(ac + b) = d (in this problem, c(c + 7) = 6). If d is not 0, you cannot conclude that c = d (or ac + b = d) (in this case c = 6 (or c + 7 = 6)). You can check that neither 6 nor 1 are solutions of c^{2} + 7c = 6. The number 0 is the only number with the property that if a product is equal to that number, then one of the factors is that number. If we had the equation c(c + 7) = 0 it would be true that c = 0 or c + 7 = 0.
There are three basic ways to solve a quadratic equation: (1) factor (the equation c^{2} + 7c  6 = 0 does not easily factor, so use a different method) (2) complete the square:
(3) use the quadratic formula: the solutions of ac^{2} + bc + d = 0 are
(see The Correction for the solution to the equation c^{2} + 7c  6 = 0 using the quadratic formula)
