Question

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

Answer 1

Let's evaluate Karim's steps to find the error in his work.

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

Here, Karim has rewritten the equation. However, the grouping he has used for the middle terms (8x and -3x) is incorrect as the coefficients should have added up to give the original coefficient of \(5x\). The correct form should reflect breaking down \(5x\) into two numbers that multiply to \(-24\) and add to \(5\), which is \(8\) and \(-3\). Thus, step 1 is incorrect.

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

This step continues from the original incorrect grouping. As step 1 was incorrect, this grouping also remains invalid.

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

This factorization is based on the incorrect groupings from Step 2.

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

The factorization assumes the previous steps were correct.

5. **Step 5**: \( x + 8 = 0 \) or \( x + 3 = 0 \)

Solutions come from Step 4, assuming it was factored correctly.

In conclusion, the first step at which Karim makes an error is Step 1 because the grouping of terms does not correctly represent the original quadratic equation. The correct breakdown of the middle term \(5x\) should yield terms that multiply to \(-24\) and add to \(5\), such as \(8\) and \(-3\). Thus, the error is in **Step 1**.