Karim is solving the quadratic equation x2+5x−24=0

Question

by grouping. His work is shown below. Determine the first step at which Karim makes an error.

Step 1: x2+8x+(−3x)−24=0

Step 2: (x2+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

(1 point)

Answer 1

Let's evaluate Karim's steps to identify where he made an error.

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

   • This step involves rewriting \( 5x \) as \( 8x - 3x \). The terms \( 8x \) and \( -3x \) do correctly add up to \( 5x \), so this step is correct.

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

   • This step groups the first two terms and the last two terms. This is also correct.

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

   • Here, Karim factors out \( x \) from the first part \( (x^2 + 8x) \) correctly, but he incorrectly factors out \( 3 \) from \( (-3x - 24) \). It should be \( -3(x + 8) \) but he mistakenly writes it as \( 3(x + 8) \).

4. Step 4: \( (x + 8)(x + 3) = 0 \)

   • This step assumes the factoring from Step 3 is correct, but since Step 3 was incorrect, this step is also incorrect.

5. Step 5: \( x + 8 = 0 \) or \( x + 3 = 0 \)

   • Even though the solutions are stated, they are based on the incorrect factors.

In summary, the first error occurs in Step 3, where Karim incorrectly factors out \( 3 \) instead of \( -3 \) from \( (-3x - 24) \). Therefore, the error occurs in Step 3.