Solving Quadratic Equations Using Common Factors Practice

To evaluate Karim's work on solving the quadratic equation \( x^2 = 5x - 24 \) by grouping, we first rearrange it into standard form:

\[
x^2 - 5x + 24 = 0
\]

Assuming Karim also rearranged it correctly, it appears he may have attempted to factor this polynomial. We'll proceed with the realization that \( x^2 - 5x + 24 \) could have been grouped (or factored) incorrectly.

Let’s look at the equation again:

1. **Standard Form**: Starting from \( x^2 - 5x + 24 = 0 \), the equation should be checked for factoring.
2. **Grouping Approach**: In grouping, one typically looks for two numbers that multiply to give the product of the coefficient of \( x^2 \) (which is 1) and the constant term (24) while adding up to \(-5\).

Now, let’s analyze the factors of \( 24 \):

- 1 and 24
- 2 and 12
- 3 and 8
- 4 and 6

None of these pairs add up to \(-5\), nor can we find two numbers that satisfy the condition for possible grouping.

Thus, at this level of analysis, if Karim has attempted to factor or group based on the expression \( x^2 - 5x + 24 \) he has made an error in several aspects:
1. The equation cannot be factored over the integers as it does not have rational roots.
2. Any manipulation to express it in a factored form like \( (x - a)(x - b) \) would have been impossible given the above observations.

The critical error would therefore be in the assumption that it could be factored by grouping when, in fact, it cannot. The first misstep occurs immediately when he tries to group the terms.

We conclude that if Karim attempted to group this quadratic for factoring, he made his error at that stage. The equation should instead be approached via the quadratic formula or by completing the square, as it does not factor cleanly.