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 $$ x=-6 $$

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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.
Elijah made his first error in Step 1 by incorrectly splitting –14x as –4x + 18x instead of –18x + 4x.

i. Elijah is solving the quadratic equation by factoring.
His work is shown below. In which step did Elijah make his first error? What
is it?

Step 1:
Step 2:
Step 3:
Step 4:
Step 5: or
or

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i. Elijah is solving the quadratic equation by factoring. His work is shown below. In which step did Elijah make his first error? What is it? Step 1: Step 2: Step 3: Step 4: Step 5: or or