Question

Karim is solving the quadratic equation x2+5x−24=0
x
2
+
5
x
−
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
x
2
+
8
x
+
(
−
3
x
)
−
24
=
0


Step 2: (x2+8x)+(−3x−24)=0
(
x
2
+
8
x
)
+
(
−
3
x
−
24
)
=
0


Step 3: x(x+8)+3(x+8)=0
x
(
x
+
8
)
+
3
(
x
+
8
)
=
0


Step 4: (x+8)(x+3)=0
(
x
+
8
)
(
x
+
3
)
=
0


Step 5: x+8=0
x
+
8
=
0
or x+3=0
x
+
3
=
0


x=−8
x
=
−
8
or x=−3
x
=
−
3


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

Answer 1

Karim's first mistake occurs at Step 1. The original equation is \(x^2 + 5x - 24 = 0\). When trying to rewrite the middle term, he incorrectly splits \(5x\) into \(8x\) and \(-3x\).

The correct grouping should involve breaking down \(5x\) into two terms that sum to \(5\) and multiply to \(-24\). For example, \(8\) and \(-3\) actually sum to \(5\), but they should also correspond to proper factorizations of the two products, which is where the problem arises.

Thus, his first mistake is indeed at Step 1.