To solve the equation \( x^2 = 2564 \) by inspection, we first take the square root of both sides.
The solutions can be expressed as:
\[ x = \pm \sqrt{2564} \]
Next, we need to simplify \( \sqrt{2564} \):
-
Factor 2564 to find its prime factors: \[ 2564 = 4 \times 641 = 2^2 \times 641 \]
-
Thus, we can simplify \( \sqrt{2564} \): \[ \sqrt{2564} = \sqrt{4 \times 641} = \sqrt{4} \times \sqrt{641} = 2\sqrt{641} \]
Now we have two solutions:
- The positive solution: \[ x = 2\sqrt{641} \]
- The negative solution: \[ x = -2\sqrt{641} \]
Finally, we write the solutions in the required format, starting with the lesser number:
- The first solution is \( x = -2\sqrt{641} \)
- The second solution is \( x = 2\sqrt{641} \)
Thus, the final answers are:
The first solution is \( x = -2\sqrt{641} \)
The second solution is \( x = 2\sqrt{641} \)