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 (1 point) Karim's first mistake occurs at Step () .

Answer 1

Karim's first mistake occurs in Step 1.

In Step 1, he incorrectly rewrites the equation \(x^2 + 5x - 24 = 0\) as \(x^2 + 8x - 3x - 24 = 0\). The correct way to rewrite the equation by grouping using two numbers that multiply to \(-24\) and add to \(5\) is to factor it correctly. The numbers that work for this are \(8\) and \(-3\) because:

- \(8 + (-3) = 5\)
- \(8 \times (-3) = -24\)

However, the way he wrote it in Step 1 specifically as \(x^2 + 8x + (-3x) - 24 = 0\) is misleading and shows an incorrect approach before any grouping.

The correct representation would have been to find two terms that correctly split \(5x\) into two parts, which leads to the next steps of grouping.

So, Karim's first mistake occurs at Step 1.