Question

Error Analysis: State which step the error was made and explain what they should have done.

Step 1: x^2 + 12x - 20 = 0
Step 2: (x^2 + 12x + _) = 20
Step 3: x^2 + 12x + 36 = 20
Step 4: (x + 6)^2 = 20
Step 5: x + 6 = ±2√5
Step 6: x = -6 ± 2√5

Answer 1

The error in solving the quadratic equation x^2 + 12x - 20 = 0 occurred in Step 3, where 36 was added without simultaneously adding it to the other side, leading to the wrong conclusion. The correct procedure involves adding 36 to both sides after step 2, then following through with factoring and solving correctly.

Step-by-step explanation:

The error in solving the quadratic equation x^2 + 12x - 20 = 0 occurred in Step 3. The mistake was adding 36 to complete the square, which was incorrect because adding b/2^2 for the equation x^2 + bx = -c would mean adding (12/2)^2, which is 36. However, since this was supposed to equal 0 originally, we should have also added 36 to the other side of the equation to maintain equality, making it x^2 + 12x + 36 = 56. This is then factored in Step 4 correctly to (x + 6)^2, but because of the error in Step 3, the right side of the equation in Step 4 should be 56, not 20.

To correct the error:

1. Right after Step 2, add 36 to both sides, giving x^2 + 12x + 36 = 56.
2. Then, in Step 4, write the equation as (x+6)^2 = 56.
3. Continue with Step 5 to find x + 6 = ±√56.
4. Lastly, solve for x in Step 6 to get the correct solutions, x = -6 ± √56, which simplifies to x = -6 ± 2√14.