i. Elijah is solving the quadratic equation \( 3 x^{2}-14 x-24=0 \) by factoring. His work is shown below. In which step did Elijah make his first error? What is it? \( 3 x^{2}-14 x-24=0 \) Step 1: \( 3 x^{2}-4 x+18 x-24=0 \) Step 2: \( \left(3 x^{2}-4 x\right)+(18 x-24)=0 \) Step 3: \( x(3 x-4)+6(3 x-4)=0 \) Step 4: \( (3 x-4)(x+6)=0 \) Step 5: \( 3 x-4=0 \) or \( x+6=0 \) \( x=\frac{4}{3} \) or \( x=-6 \)

Elijah's approach involves factoring the quadratic equation \( 3x^2 - 14x - 24 = 0 \) by splitting the middle term. Let's examine each step to identify where the error occurs. ### Step-by-Step Analysis: 1. **Original Equation:** \[ 3x^2 - 14x - 24 = 0 \] 2. **Step 1: Splitting the Middle Term** \[ 3x^2 - 4x + 18x - 24 = 0 \] **Intended Method:** - For a quadratic equation \( ax^2 + bx + c \), find two numbers \( m \) and \( n \) such that: \[ m \times n = a \times c = 3 \times (-24) = -72 \] \[ m + n = b = -14 \] - The correct numbers should be \( m = -18 \) and \( n = 4 \) because: \[ -18 \times 4 = -72 \quad \text{and} \quad -18 + 4 = -14 \] - Therefore, the middle term should be split as \( -18x + 4x \), not \( -4x + 18x \). 3. **Error Identification:** - **Error Location:** **Step 1** - **Nature of Error:** Elijah incorrectly split the middle term \( -14x \) as \( -4x + 18x \) instead of the correct \( -18x + 4x \). - This incorrect splitting disrupts the subsequent factoring steps, leading to an incorrect factorization and wrong solutions. 4. **Correct Approach:** - **Correct Step 1:** \[ 3x^2 - 18x + 4x - 24 = 0 \] - **Continuing Correctly:** \[ (3x^2 - 18x) + (4x - 24) = 0 \] \[ 3x(x - 6) + 4(x - 6) = 0 \] \[ (3x + 4)(x - 6) = 0 \] \[ x = -\frac{4}{3} \quad \text{or} \quad x = 6 \] ### Conclusion: **Elijah made his first error in Step 1 by incorrectly splitting the middle term \( -14x \) as \( -4x + 18x \) instead of the correct \( -18x + 4x \).** **Answer:** Elijah erred in Step 1 by incorrectly splitting –14x as –4x + 18x instead of –18x + 4x.