a. Watch this video of Mrs. Hool completing a similar problem b. Reattempt the question from your test: 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 1:3x 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 \( x=6 \)

To determine where Elijah made his first error in solving the quadratic equation \( 3 x^{2}-14 x-24=0 \) by factoring, let's analyze each step of his work. 1. **Original Equation**: \[ 3 x^{2}-14 x-24=0 \] 2. **Step 1**: Elijah writes \( 3x \). This step is unclear and does not provide any useful information. 3. **Step 2**: \[ (3 x^{2}-4 x)+(18 x-24)=0 \] Here, Elijah attempts to group the terms. However, the grouping is incorrect. The correct grouping should maintain the original equation's structure. The term \( -14x \) should be split into two terms that add up to \( -14x \) while also considering the product of the leading coefficient and the constant term. 4. **Step 3**: \[ x(3 x-4)+6(3 x-4)=0 \] This step is based on the incorrect grouping from Step 2. Therefore, it is also incorrect. 5. **Step 4**: \[ (3 x-4)(x+6)=0 \] This step is derived from the previous incorrect step. 6. **Step 5**: \[ 3 x-4=0 \quad \text{or} \quad x+6=0 \] This leads to the solutions \( x=\frac{4}{3} \) or \( x=6 \). ### Conclusion: Elijah's first error occurred in **Step 2**, where he incorrectly grouped the terms. The correct approach would involve factoring the quadratic expression directly or using the quadratic formula. To find the correct factors, we can use the method of factoring directly from the original equation: 1. The product of \( a \) (3) and \( c \) (-24) is \( -72 \). 2. We need two numbers that multiply to \( -72 \) and add to \( -14 \). The correct pair is \( -18 \) and \( 4 \). 3. Thus, we can rewrite the equation as: \[ 3x^2 - 18x + 4x - 24 = 0 \] 4. Factoring gives: \[ 3x(x - 6) + 4(x - 6) = 0 \] 5. This leads to: \[ (3x + 4)(x - 6) = 0 \] The correct solutions would then be: \[ 3x + 4 = 0 \quad \Rightarrow \quad x = -\frac{4}{3} \] \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] Thus, Elijah's first error was in Step 2, where he incorrectly grouped the terms.