8.

Question

Solve x2= 2564
by inspection. There are 2 real solutions. Enter the lesser number first. Leave the answers in simplest fraction form.

(1 point)
The first solution is x=

The second solution is x =

Answer 1

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 \( \sqrt{2564} \):

1. Factor 2564 to find its prime factors: \[ 2564 = 4 \times 641 = 2^2 \times 641 \]

2. 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:

1. The first solution is \( x = -2\sqrt{641} \)
2. 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 \( x = 2\sqrt{641} \)