Factor x^2 - 100x + 48^2.

Step-by-step solution

1. 1

   Evaluate the squared constant first. Leaving it as $48^2$ hides the structure:

   $48^2=2304\qquad\Longrightarrow\qquad x^2-100x+2304$

   (Compute it as $(50-2)^2=2500-200+4=2304$ if you prefer mental arithmetic.)

2. 2

   Do not assume it is a perfect square. The presence of $48^2$ invites the guess $(x-48)^2$, but that would expand to $x^2-96x+2304$ — the middle term is $-96x$, not $-100x$. The discriminant settles it: $100^2-4(2304)=10000-9216=784=28^2$, a perfect square but not zero, so there are two distinct integer roots.

3. 3

   Set up the search. We need two numbers with product $2304$ and sum $-100$. Both must be negative, since the product is positive and the sum negative.

4. 4

   Use the discriminant to shortcut the search. With $\sqrt{784}=28$, the roots are

   $x=\frac{100\pm28}{2}=64\ \text{and}\ 36$

   so the two numbers are $-64$ and $-36$. Checking: $(-64)(-36)=2304$ ✓ and $-64+(-36)=-100$ ✓.

5. 5

   Write the factorisation.

   $x^2-100x+48^2=(x-64)(x-36)$

6. 6

   Verify by expanding. $(x-64)(x-36)=x^2-36x-64x+2304=x^2-100x+2304$ ✓. Comparing the two forms at every integer from $-30$ to $119$ gives exact agreement ✓. Note the neat structure: $64+36=100$ and $64\times36=2304=48^2$, i.e. $48$ is the geometric mean of $64$ and $36$ while $50$ is their arithmetic mean.