Karim is solving the quadratic equation x2+5x−24=0

Question

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 factors the quadratic expression \(x^2 + 5x - 24\) as \(x^2 + 8x + (-3x) - 24 = 0\). The correct grouping to factor this equation should involve finding two numbers that multiply to \(-24\) and add to \(5\). The correct pair would be \(8\) and \(-3\), resulting in the correct factorization being:

\[ x^2 + 8x - 3x - 24 = 0 \]

However, the grouping using positive \(8\) and negative \(3\) should directly compare \(5\) instead, which doesn't appear in the expression he attempted to create. Since he doesn't correctly decompose the middle term of \(5x\), this is where the error first takes place.