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

Question

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

Answer 1

Let's analyze Karim's steps in solving the quadratic equation \( x^2 + 5x - 24 = 0 \) by grouping.

Step 1: \( x^2 + 8x + (-3x) - 24 = 0 \)

In this step, Karim attempts to rewrite the middle term (5x) as the sum of two terms (8x and -3x). However, this is where the error occurs.

The correct split of the middle term (5x) should maintain the equality and should sum to 5. The right approach would be to factor two numbers that multiply to -24 (the product from \( a \cdot c \) where \( a = 1 \) and \( c = -24 \)) and add to 5. The correct numbers that work here are +8 and -3, which would look like this:

Step 1 should have been: \( x^2 + 8x - 3x - 24 = 0 \)

While his expression can be rearranged and factored, the incorrect naming of it as \( x^2 + 8x - 3x - 24 \) led to him misrepresenting the equation to start with.

Therefore, the first error occurs in Step 1 where he incorrectly rewrites the equation. The numbers cannot simply be grouped without maintaining their original values correctly. The proper grouping starts with the correct middle term expression.