Find the error in each of the following solutions. Explain your reasoning. (6 marks)
a) 2š„
2 ā 12š„ ā 21 = 0
š„
2 ā 6š„ =
21
2
š„
2 ā 6š„ + 9 =
21
2
ā 9
(š„ ā 3)
2 =
3
2
š„ ā 3 = Ā±ā
3
2
š„ ā 3 = Ā±
ā3
ā2
š„ = 3 Ā±
ā6
2
The error in this solution is at the step where they square both sides of the equation (š„-3)^2 = 3/2. When squaring both sides of an equation, we need to consider the possibility of introducing extraneous solutions. In this case, when squaring both sides, we also introduce the solution š„=3, which is not a solution to the original equation. Therefore, the step of squaring both sides is incorrect.
The correct steps would be:
1) Start with the equation 2š„^2 - 12š„ - 21 = 0.
2) Factor the equation: (š„-3)(2š„+7) = 0.
3) Set each factor equal to zero and solve for š„:
š„-3 = 0 ā š„ = 3
2š„+7 = 0 ā š„ = -7/2.
4) The solutions to the equation are š„ = 3 and š„ = -7/2.