The hypotenuse of a right triangle is 8m long. One leg is 2m longer then the other. Find the lengths of the legs. How would you solve this equation?

short leg --- x

longer leg --- x+2

x^2 + (x+2)^2 = 8^2
x^2 + x^2 + 4x + 4 - 64 = 0
2x^2 + 4x - 60 = 0
x^2 + 2x - 30 = 0
let's complete the square:
x^2 + 2x + 1 = 30+1 = 31
(x+1)^2 = 31
x+1 = ± √31
x = -1 ± √31
but x has to be positive,
so x = √31 - 1 = appr 4.568

so one side is 4.568, the other is 6.568

check:
4.568^2 + 6.568^2 = 64.0052
close enough to 64

To solve this equation, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's assume that one leg of the right triangle is x meters long. Since the other leg is 2 meters longer, its length will be x + 2.

Using the Pythagorean theorem, we can write the equation:

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

Simplifying this equation, we have:

x^2 + (x^2 + 4x + 4) = 64

Combining like terms:

2x^2 + 4x + 4 = 64

Rearranging the equation and simplifying:

2x^2 + 4x - 60 = 0

Now we have a quadratic equation. We can solve this equation by factoring or using the quadratic formula:

Factoring:

2(x^2 + 2x - 30) = 0

(x + 6)(x - 5) = 0

So, x + 6 = 0 or x - 5 = 0

This gives us two possible solutions for x: x = -6 or x = 5.

However, since we are dealing with lengths, we can disregard the negative value. Thus, x = 5.

Therefore, one leg of the right triangle is 5 meters long, and the other leg will be 2 meters longer, which is 7 meters.

To solve this equation, we can use the Pythagorean theorem, which states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

Let's assume the lengths of the legs are x and x+2 (since one leg is 2 meters longer than the other).

According to the Pythagorean theorem, we have the following equation:

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

Simplifying the equation, we have:

x^2 + x^2 + 4x + 4 = 64

Combining like terms, we get:

2x^2 + 4x + 4 = 64

Moving all terms to one side of the equation, we get:

2x^2 + 4x - 60 = 0

Now, to solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

In this equation, a = 2, b = 4, and c = -60.

Plugging these values into the quadratic formula, we get:

x = (-4 ± √(4^2 - 4(2)(-60)))/(2(2))

Simplifying further, we have:

x = (-4 ± √(16 + 480))/4

x = (-4 ± √496)/4

Now, let's simplify the square root:

x = (-4 ± √(16 * 31))/4

x = (-4 ± 4√31)/4

Now, let's simplify further by factoring out a 4 in the numerator:

x = (4(-1 ± √31))/4

Simplifying the fraction, we have:

x = -1 ± √31

Therefore, the possible values for x (the length of the shorter leg) are approximately -1 + √31 and -1 - √31.

Since we are dealing with lengths, the value of x cannot be negative. Therefore, the length of the shorter leg is approximately -1 + √31.

To find the length of the longer leg, we can use the equation x+2. Plugging the value of x, we have:

Length of the shorter leg = -1 + √31
Length of the longer leg = (-1 + √31) + 2

Simplifying further, we have:

Length of the longer leg ≈ 1 + √31

So, the lengths of the legs of the right triangle are approximately -1 + √31 meters and 1 + √31 meters.