Question

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

Answer 1

Let's analyze Karim's steps in solving the quadratic equation \( x^2 + 5x - 24 = 0 \) by grouping to find the error.

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

In this step, Karim rewrites the middle term \( 5x \) as \( 8x - 3x \). The decomposition is incorrect because:
\[ 8 + (-3) \neq 5 \]
Since \( 8 - 3 = 5 \) looks correct in coefficient terms here, it will help determine the validity of the next steps.

However, the proper coefficient split for \( 5x \) should yield two numbers whose product is \(-24\) and sum is \(5\), which would be \(8\) and \(-3\) for \(x^2 + 5x - 24\). This means he does the correct split.

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

This grouping is acceptable based on Step 1.

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

Here, he factors each group. However, there is a slight inconsistency since:
The proper factoring would have taken:
- From \( -3x - 24\), factoring out -3 gives:
\(-3(x + 8)\) but it should be:
\(-(3)(x + 8)\) would have been applicable since adjusting for signs must show clearly.

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

Step 4 misleads directly given that it's not clear how the negative units distribute across common terms from above.

**Final Step Analysis:**
When he factors out \((x + 8)\) out correctly, but mismatches the required signs from step grouping earlier. The follow-up analysis of factors yields \(x + 3 = 0\) results visually correct since it concerns resolution proper handling of \( x + (-8)\), hence causing a sign problem.

It seems the misalignment begins at:

**Step 2** or Step 3, as they failed from exact structure followed up. \( \textbf{Step 1}\) has a critical path leading misunderstanding on resulting factors concludes into a wrong solution overall after further analysis.

Thus, the first major point of discrepancy occurs when trying the factorization in grouping leading to the errors forward since extraction/factorization would yield misaligned signs handlings of signs in arrangement.

Finalizing Steps, the reliable initial step causing trouble occurs thus efficiently from \( 3^{rd} \) leading misresult resolved to positives needed thus \( (x + 3) \) pulling through leading to reverses on negatives hence yielding negative as primary stated earlier. Thus, the misstep is aggregately at:
**Our step identification pivots at ensuring \( x^2 + 8x (error remainder would yield negative resolutions).**