Solve the equation x^2 - 25 = 0 by factoring.

Step-by-step solution

1. 1

   Spot the pattern before choosing a method. There is no $x$ term, and both $x^2$ and $25$ are perfect squares with a minus sign between them. That is the signature of a difference of two squares:

   $A^2-B^2=(A-B)(A+B)$

2. 2

   Identify $A$ and $B$. Here $A=x$ because $x^2=(x)^2$, and $B=5$ because $25=5^2$. Substituting:

   $x^2-25=(x-5)(x+5)$

3. 3

   Set the factored form to zero and split.

   $(x-5)(x+5)=0\ \Longrightarrow\ x-5=0\ \text{or}\ x+5=0$

4. 4

   Solve both branches.

   $x=5\qquad\text{or}\qquad x=-5$

   The factoring route makes the negative root impossible to forget. Going straight to $x^2=25$ and writing $x=\sqrt{25}=5$ loses the second root — the correct statement there is $x=\pm\sqrt{25}$.

5. 5

   Check both values.

   $5^2-25=25-25=0\quad\checkmark$

   $(-5)^2-25=25-25=0\quad\checkmark$