Question

Karim is solving the quadratic equation x^2 +5x -24=0 by grouping his work is shown below. Determine the first step at which he makes an error
Step1. x^2 + 8x + (-3x) - 24 = 0
Step 2. (x^2 +8x) + (-3x -24)=0
Step 3. x(x +8) + 3(x+ 8)= 0
Step 4. (x + 8)(x + 3) =0
Step 5. x +8 = 0 or x + 3 = 0
x= -8 or x = -3

Answer 1

To determine the first step at which Karim makes an error, let's analyze each step of his work:

1. **Step 1:** \( x^2 + 8x + (-3x) - 24 = 0 \)

Here, Karim is attempting to rewrite the middle term of the quadratic equation. However, he should be looking for two numbers that add up to \(5\) (the coefficient of \(x\)) and multiply to \(-24\) (the constant term). The factors should be \(8\) and \(-3\) if he is grouping, but let's check the grouping process.

2. **Step 2:** \( (x^2 + 8x) + (-3x - 24) = 0 \)

This step is valid.

3. **Step 3:** \( x(x + 8) + 3(x + 8) = 0 \)

Here, the grouping is incorrect because if he groups \(x^2 + 8x\) together, he gets \(x(x + 8)\), but he should be grouping \(-3x - 24\) as such: \(-3(x + 8)\). However, he mistakenly factored it as \(3(x + 8)\), which is incorrect because of the negative sign in front of \(3x\).

4. **Step 4:** From the incorrect factorization in Step 3, this leads to \( (x + 8)(x + 3) = 0 \).

5. **Step 5:** He then solves for \(x\), yielding \(x + 8 = 0\) or \(x + 3 = 0\), leading to \(x = -8\) or \(x = -3\).

**Conclusion:**
The first error occurs in Step 3, where he incorrectly factors \(-3x - 24\) as \(3(x + 8)\) instead of \(-3(x + 8)\). Thus, the correct route would have been \(x^2 + 5x - 24 = 0\) directly to \( (x + 8)(x - 3) = 0 \).