A piece of wire is 76cm long.it is cut into two unequal parts and each part is bent into a square.the sum of the areas of two squares is 205cm^2.find the length of the shorter part of the wire.

[(76 - x) / 4]² + (x / 4)² = 205

76² - 152 x + x² + x² = 205 * 16

use the quadratic formula to find x

either x or 76-x is shorter

Let's assume that the length of the longer part of the wire is x cm, therefore the length of the shorter part would be (76 - x) cm.

When a wire is bent into a square, each side of the square would be equal to one-fourth the length of the wire. Thus, the side length of the square made from the longer part would be (x/4) cm, and the side length of the square made from the shorter part would be ((76 - x)/4) cm.

To find the area of a square, we square the length of its side. Therefore, the area of the square made from the longer part would be (x/4)^2 cm^2, and the area of the square made from the shorter part would be ((76 - x)/4)^2 cm^2.

According to the problem, the sum of the areas of the two squares is 205 cm^2. Therefore, we can write the equation:

(x/4)^2 + ((76 - x)/4)^2 = 205

Simplifying, we get:

x^2/16 + (5776 - 152x + x^2)/16 = 205

Combining the terms, we have:

x^2 + 5776 - 152x + x^2 = 3280

Simplifying further:

2x^2 - 152x + 2496 = 0

Dividing the equation by 2, we get:

x^2 - 76x + 1248 = 0

Now let's solve this quadratic equation using factoring or quadratic formula:

(x - 48)(x - 26) = 0

This gives us two possible values for x: x = 48 or x = 26.

Since we assumed x to be the length of the longer part, we can conclude that the length of the shorter part is (76 - x), and thus, it is equal to 76 - 48 = 28 cm.

So, the length of the shorter part of the wire is 28 cm.

To solve this problem, we need to set up some equations based on the given information.

Let's assume the length of the shorter part of the wire is x cm. Then the length of the longer part would be (76 - x) cm.

Since both parts are bent into squares, we can find the sides of the squares using the given lengths.

The formula for the perimeter of a square is P = 4s, where P represents the perimeter and s represents the side length.

For the shorter part, the perimeter would be equal to the length of the wire, which is x cm:
4s1 = x

For the longer part, the perimeter would be equal to the length of the wire minus the length of the shorter part:
4s2 = (76 - x)

Now, let's find the areas of the two squares. The formula for the area of a square is A = s^2, where A represents the area and s represents the side length.

The sum of the areas of the two squares is given as 205 cm^2:
s1^2 + s2^2 = 205

To solve for x, we can substitute the values of s1 and s2 from the perimeter equations into the area equation.

Substituting s1 = x/4 and s2 = (76 - x)/4 into the area equation, we get:
(x/4)^2 + ((76 - x)/4)^2 = 205

Now we can solve this quadratic equation to find x.