Question

Karim is solving the quadratic equation x2+5x−24=0
by grouping. His work is shown below. Determine the first step at which Karim makes an error.

Step 1: x2+8x+(−3x)−24=0


Step 2: (x2+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 rewrote the original equation \(x^2 + 5x - 24 = 0\) as \(x^2 + 8x - 3x - 24 = 0\). However, the coefficients must sum to the original coefficient of \(x\), which is 5. The expression \(8x - 3x\) simplifies to \(5x\), but the number 8 was incorrectly chosen.

The correct approach at Step 1 should have involved finding two numbers that multiply to -24 (the constant term) and add to 5 (the coefficient of \(x\)). The correct numbers are 8 and -3, but the grouping in the context is incorrect as he mistakenly decided to separate them incorrectly.

So the first mistake occurs at Step 1.