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)
Karim's first mistake occurs at Step
.
Remaining Attempts : 3

Answer 1

Let's analyze Karim's steps to determine where he made an error.

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

   The original equation is \( x^2 + 5x - 24 = 0 \). The correct split for \( 5x \) using grouping would be \( 8x + (-3x) \), which is correct because \( 8 + (-3) = 5 \). So, this step is valid.

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

   In this step, Karim has grouped his terms correctly. \( (x^2 + 8x) \) and \( (-3x - 24) \) are indeed correctly grouped. So, this step is also valid.

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

   Here, he factored out \( x \) from the first grouped term and \( 3 \) from the second. However, he factored \( -3 \) instead of \( 3 \) since he was extracting the factors from \( -3x - 24 \), which should have factored as \( -3(x + 8) \). So this step has an error.

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

   Since his previous step had an error, this step also ends up being incorrect.

5. Step 5:

   This step is based on the incorrect factored expression from the previous step and thus produces incorrect solutions.

In conclusion, Karim's first mistake occurs at Step 3.