Hyung-eun is solving the quadratic equation x^2 - 4x -12 = 0 by grouping. Her work is shown below. At whic step does Hyung-eun first make an error?
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
A. Step 3
B. Step 1
C. Step 4
D. Step 2
The error occurs in Step 3.
The correct factorization should be x(x - 6) + 2(x - 6) = 0, but Hyung-eun mistakenly wrote it as x(x - 6) + 2(x - 6) = 0.
The correct equation should be x(x - 6) + 2(x - 6) = 0.
The correct answer is D. Step 2.
In Step 2, Hyung-eun incorrectly grouped the terms as (x^2 + (-6x)) + (2x - 12), which should have been (x^2 - 6x) + (2x - 12).
This mistake changes the signs in the equation and leads to an error in Step 3.
Step 3: x (x + (-6)) + 2(x - 6) = 0 is where Hyung-eun first makes an error.
To solve the quadratic equation x^2 - 4x - 12 = 0 by grouping, we need to factorize the quadratic expression. The correct steps would be:
Step 1: x^2 - 4x - 12 = 0
Step 2: Rearrange the equation as x^2 + (-6x) + 2x - 12 = 0, so we can group terms.
Step 3: Grouping the terms, we have (x^2 + (-6x)) + (2x - 12) = 0.
Step 4: Factor out the common factors in each group, which results in x(x - 6) + 2(x - 6) = 0. Here, both terms have a factor of (x - 6).
Step 5: Combine like terms, we get (x - 6)(x + 2) = 0.
Step 6: Apply the zero product property, which states that if the product of two terms equals zero, then at least one of the terms must be zero. Thus, we have two equations: (x - 6) = 0 or (x + 2) = 0.
Step 7: Solve each equation separately. From (x - 6) = 0, we find x = 6. From (x + 2) = 0, we find x = -2.
By analyzing the provided steps, we can see that in Step 3, Hyung-eun incorrectly distributed the 2 to both terms in the parentheses, resulting in x(x - 6) + 2(x - 6) = 0 instead of the correct expression x(x - 6) + 2(x - 6) = 0. Therefore, the answer is A. Step 3.