Karim is solving the quadratic equation x^2 + 5x - 24 = 0 by grouping his work is shown below determine the first step at which karim makes an error

Question

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
Which step did karim make her first mistake

Answer 1

Karim made his first mistake in Step 1.

In the original equation \( x^2 + 5x - 24 = 0 \), he attempted to split the middle term to factor by grouping. However, he incorrectly wrote \( 8x \) and \( -3x \) as the terms for the split. The correct way to split \( 5x \) into two terms that would allow for factoring is not \( 8x \) and \( -3x \).

To find two numbers, \( a \) and \( b \), that multiply to \( -24 \) (the constant term) and add up to \( 5 \) (the coefficient of the \( x \) term), we actually need \( 8 \) and \( -3 \), as he later stated, but this was never clear from step 1, and the split was incorrect.

Thus, the first mistake was in Step 1, where the expression should be split as \( 8x + (-3x) \). The more proper terms should relate better to creating factors of the quadratic equation, as we see later in his steps the factors do not align with the original quadratic.