Question

Find the error in each of the following solutions. Explain your reasoning. (6 marks)
a) 2𝑥
2 − 12𝑥 − 21 = 0
𝑥
2 − 6𝑥 =
21
2
𝑥
2 − 6𝑥 + 9 =
21
2
− 9
(𝑥 − 3)
2 =
3
2
𝑥 − 3 = ±√
3
2
𝑥 − 3 = ±
√3
√2
𝑥 = 3 ±
√6
2

Answer 1

The error in this solution is at the step where they square both sides of the equation (𝑥-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 𝑥=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 2𝑥^2 - 12𝑥 - 21 = 0.
2) Factor the equation: (𝑥-3)(2𝑥+7) = 0.
3) Set each factor equal to zero and solve for 𝑥:
𝑥-3 = 0 → 𝑥 = 3
2𝑥+7 = 0 → 𝑥 = -7/2.
4) The solutions to the equation are 𝑥 = 3 and 𝑥 = -7/2.