Let's analyze Karim's work step by step to identify where he made an error.
-
Step 1: Karim rewrites the equation \(x^2 + 5x - 24 = 0\) as \(x^2 + 8x + (-3x) - 24 = 0\).
- The error here is that \(8 - 3\) does not equal \(5\). The correct way to break down the \(5x\) term would be to find two numbers that multiply to \(-24\) (the constant term) and add up to \(5\). The two numbers that work are \(8\) and \(-3\), but this should be expressed correctly.
-
Step 2: He groups the terms as \((x^2 + 8x) + (-3x - 24) = 0\).
- This grouping is not valid since the first step itself was incorrect. It should have been \(x^2 + 8x - 3x - 24\).
-
Step 3: He factorizes this expression as \(x(x + 8) + 3(x + 8) = 0\).
- This step seems to follow from the previous incorrect step, which means if his previous step was wrong, so is this one.
-
Step 4: He factors it to \((x + 8)(x + 3) = 0\).
- This also follows from the incorrect factorization of the previous steps.
-
Step 5: Finally, he finds the roots \(x + 8 = 0\) or \(x + 3 = 0\), giving \(x = -8\) or \(x = -3\).
- These roots would be valid if the previous steps were valid, but since they stem from an earlier mistake, they are incorrect.
Conclusion: The first error occurs in Step 1, where he incorrectly rewrites the equation and splits the \(5x\) into \(8x\) and \(-3x\). The correct approach should not have added \(8x\) in the first place. He needed to adequately break down \(5x\) as \(8x - 3x\) but following the proper structure to lead towards factoring.