Karim is solving the quadratic equation x^2+5x−24=0

Question

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

Step 1: 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

(1 point)
Karim's first mistake occurs at Step ___.

Answer 1

Karim's first mistake occurs at Step 1.

In Step 1, he rewrites the quadratic equation \(x^2 + 5x - 24 = 0\) incorrectly as \(x^2 + 8x + (-3x) - 24 = 0\). The correct way to group it would involve splitting the middle term \(5x\) into two terms that add up to \(5x\) and multiply to give the product of the leading coefficient (\(1\)) and the constant term (\(-24\)).

The correct factorization of the middle term could be \(8x - 3x\), which would not yield the correct sum. In fact, \(8x - 3x\) gives \(5x\), but it doesn't correspond to the correct factors for \(x^2 + 5x - 24\).

Thus, Karim's error is in Step 1.

Answer 2

Thats wrong

Answer 3

I apologize for the confusion in my previous response. Let's carefully go through Karim's steps to identify the error.

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

   This splits the middle term incorrectly since \(8 + (-3) = 5\) but \(8 \times (-3) \neq -24\). This does not satisfy what we need.

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

   This is a correct grouping based on his previous split.

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

   This is also a correct factoring based on the previous step.

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

   This factoring is correct given the previous step.

5. Step 5: Solving gives \(x + 8 = 0\) or \(x + 3 = 0\) leading to the solutions:

   • \(x = -8\)
   • \(x = -3\)

The error actually occurs specifically at Step 1, where he incorrectly rewrote the equation \(x^2 + 5x - 24 = 0\) as \(x^2 + 8x - 3x - 24 = 0\). The valid pairs that would satisfy \(5x\) would require different coefficients.

Thus, the answer remains that Karim's first mistake occurs at Step 1. If I misunderstood your clarification or if you believe the error lies elsewhere, please let me know!