Hyung-eun is solving the quadratic equation x^2 - 4x - 12 = 0. by grouping. Her work is shown below. At which step does Hyung-eun first make an error?

Question

x^2 - 4x - 12 = 0
Step 1: x^2 + (-6x) + 2x - 12 = 0
Step 2: (x^2 + (-6x)) + (2x - 12) = 0
Step 3: x (x + (-6)) + 2 (x - 6) = 0
x (x - 6) + 2 (x - 6) = 0
Step 4: (x + 6) (x - 2) = 0
Step 5: x + 6 = 0 or x - 2 = 0
x = -6 or x = 2

Answer 1

Let's analyze Hyung-eun's steps to see where the error occurred.

1. Step 1: Hyung-eun rewrites the equation as \( x^2 + (-6x) + 2x - 12 = 0 \).

   • Here, the original equation \( x^2 - 4x - 12 = 0 \) was manipulated incorrectly. The correct way to express -4x, which is the middle term, would involve breaking it down into two terms whose sum is -4 and whose product is -12 (the constant term). For example, -6 and +2 are not correct pairs to split -4 because they have a sum of -4 but will lead to incorrect grouping.

2. Step 2: (x^2 + (-6x)) + (2x - 12) = 0.

   • This step follows directly from Step 1, so it carries through the error.

3. Step 3: x(x + (-6)) + 2(x - 6) = 0

   • In this step, the factored terms from Step 2 are continued along but the factors themselves are based on the already incorrect analysis from Step 1.

4. Step 4: (x + 6)(x - 2) = 0

   • This step contains an error based on the previous ones. The correct factorization (which was not achieved) should have related to (x - 6)(x + 2), stemming from the quadratic equation being factored correctly.

5. Step 5: x + 6 = 0 or x - 2 = 0

   • The solutions here are incorrect based on the wrong factorization in earlier steps.

Conclusion: The first error occurs in Step 1 when Hyung-eun incorrectly splits the middle term -4x into -6x + 2x. The accurate approach requires finding two numbers that add up to -4 and multiply to -12, which would be -6 and +2.