The side of retangular floor are xm and (x + 7)m.the diagonal is (x + 8)m,calculate the value of x

using phythagoras theorm

(x+8)^2=x^2+(x+7)^2
x^2+16x+64=x^2+x^2+14x+49
16x+64=x^2+14x+49
x^2+14x-16x+49-64=0
x^2-2x-15=0
(x+3)(x-5)=0
x=5........

To find the value of x, we can use the Pythagorean theorem because the diagonal of a rectangle forms a right triangle with the sides of the rectangle.

According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.

In this case, the hypotenuse (diagonal) is (x + 8)m, and the other two sides are xm and (x + 7)m.

Therefore, we can set up the equation as follows:

(x + 8)^2 = x^2 + (x + 7)^2

Expanding both sides of the equation:

x^2 + 16x + 64 = x^2 + x^2 + 14x + 49

Simplifying:

x^2 + 16x + 64 = 2x^2 + 14x + 49

Rearranging the terms:

2x^2 + 14x + 49 - x^2 - 16x - 64 = 0

Combining like terms:

x^2 - 2x - 15 = 0

Now we can solve this quadratic equation. Factoring or using the quadratic formula, we find:

(x - 5)(x + 3) = 0

Setting each factor equal to zero:

x - 5 = 0 or x + 3 = 0

Solving for x:

x = 5 or x = -3

Since we are dealing with the length of sides, the value of x cannot be negative. Therefore, x = 5 is the valid solution.

So, the value of x is 5 meters.

To calculate the value of x, we can use the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the rectangular floor can be seen as a right-angled triangle, with the length x and x+7 as the two shorter sides, and the length x+8 as the hypotenuse.

So, applying the Pythagorean Theorem, we have the equation:

(x^2) + ((x+7)^2) = (x+8)^2

Expanding and simplifying the equation, we get:

x^2 + (x^2 + 14x + 49) = x^2 + 16x + 64

Combining like terms, we have:

2x^2 + 14x + 49 = x^2 + 16x + 64

Rearranging the equation to isolate the x terms, we have:

2x^2 + 14x - x^2 - 16x = 64 - 49

Simplifying further, we get:

x^2 - 2x - 15 = 0

Now, we can factor this quadratic equation or use the quadratic formula to find the values of x. Factoring this equation, we have:

(x - 5)(x + 3) = 0

Setting each factor equal to zero, we get:

x - 5 = 0 or x + 3 = 0

Solving for x, we have:

x = 5 or x = -3

However, in this case, we can ignore the negative value of x since we are dealing with the length of a side, which cannot have a negative value.

Therefore, the value of x is 5.