Question

- 8/w-2=- 5/4w-8+2

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve this equation, let's first clear the denominators by multiplying every term by their respective denominators.

Starting with the left side of the equation:
-8/w - 2 = -5/4w - 8 + 2.

We can rewrite -8/w as -8 * (1/w) = -8w^-1.

Now, multiplying every term by w to clear the denominator, we get:
-8 - 2w = -5/4w^2 - 8w + 2w.

Next, we will combine like terms on both sides of the equation.
On the right side, combining -8w and 2w, we get -6w.
On the left side:
-8 - 2w = -8w^2/4 - 6w.

Simplifying the equation, we now have:
-2w - 8 = -2w^2 - 6w.

Next, let's move all the terms to one side of the equation:
-2w + 2w^2 - 6w - 8 = 0.

Adding 8 to both sides:
-2w + 2w^2 - 6w = 8.

Combining like terms on the left side:
2w^2 - 8w = 8.

Now, let's rearrange the equation in descending order:
2w^2 - 8w - 8 = 0.

Next, we can divide the entire equation by 2 to simplify it:
w^2 - 4w - 4 = 0.

To solve this quadratic equation, we can use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / 2a.

In this case, a = 1, b = -4, and c = -4.

Plugging these values into the quadratic formula, we get:
w = (-(-4) ± √((-4)^2 - 4(1)(-4))) / 2(1),
w = (4 ± √(16 + 16)) / 2,
w = (4 ± √(32)) / 2,
w = (4 ± 4√2) / 2.

Simplifying further:
w = (2(2 ± 2√2)) / 2,
w = (2 ± 2√2).

Therefore, the solutions to the equation are w = 2 + 2√2 and w = 2 - 2√2.