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

Question

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 at Step 1.

In Step 1, he incorrectly rewrote the quadratic equation \(x^2 + 5x - 24 = 0\) as \(x^2 + 8x + (-3x) - 24 = 0\). The correct grouping should involve splitting the middle term, \(5x\), in a way that the two terms sum to \(5x\) and multiply to \(-24\).

To factor \(x^2 + 5x - 24\) correctly, we need to find two numbers that add to \(5\) and multiply to \(-24\). The correct numbers would be \(8\) and \(-3\), leading to \(x^2 + 8x - 3x - 24 = 0\), but he incorrectly grouped them first. Thus, the error occurs here.

So, the answer is: Step 1.