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.

In Step 1, he incorrectly rewrote the middle term \(5x\) as \(8x + (-3x)\). The correct way to split \(5x\) for factoring would be to find two numbers that multiply to \(-24\) (the product of the constant term and the coefficient of \(x^2\)) and add to \(5\). The correct factorization should have been \(8\) and \(-3\) as the terms make up \(5\); however, the error lies in how he treats these numbers, as \(5x\) cannot be represented as \(8x + (-3x)\) accurately in what he is trying to accomplish.

The correct first step would have been finding two numbers that multiply to \(-24\) and add to \(5\), which are \(8\) and \(-3\), leading to a correct formulation for grouping and factoring.

Thus, the first error is in Step 1.