Solve: 5/(x-8)-5/x = 40/(x^2-8x)
How would this be solved?
Multiply both sides of the equation by
x(x-8)
Please stop changing posting names. It does you no good when working with volunteers when you try to trick them. It is difficult enough to work through these with students who are stuck in their thinking, but much more so when those students pretend to be new at the work.
Thanks.
x^2-8x = x(x-8) which is LCD
so
5(x) - 5(x-8) = 40
x - x + 8 = 5
There is no solution. x can be anything
Thank You Damon I understand now.
To solve the equation 5/(x-8) - 5/x = 40/(x^2 - 8x), follow these steps:
1. Start by finding a common denominator for all three fractions. The denominators are (x-8), x, and (x^2-8x). The least common denominator (LCD) should include all these factors. In this case, the LCD is x(x-8).
2. Multiply the entire equation by the LCD, so each fraction's denominator becomes x(x-8). This step eliminates the fractions.
x(x-8) * [5/(x-8) - 5/x] = x(x-8) * 40/(x^2 - 8x)
Simplifying the left side gives: 5x - 5(x-8), which reduces to 40/(x^2 - 8x)
Multiplying the right side gives: 5x(x-8)
The equation becomes: 5x - 5(x-8) = 5x(x-8)
3. Simplify both sides of the equation:
5x - 5x + 40 = 5x^2 - 40x
40 = 5x^2 - 40x
4. Rearrange the equation into quadratic form, so that one side is zero:
5x^2 - 40x - 40 = 0
5. To solve the quadratic equation, you can either factor it or use the quadratic formula. This equation is not easily factorable, so use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 5, b = -40, and c = -40.
Plugging these values into the quadratic formula gives:
x = (-(-40) ± √((-40)^2 - 4 * 5 * -40)) / (2 * 5)
Simplifying gives: x = (40 ± √(1600 + 800)) / 10
x = (40 ± √(2400)) / 10
x = (40 ± √(400 * 6)) / 10
x = (40 ± 20√6) / 10
Simplifying further gives the two solutions:
x₁ = (40 + 20√6) / 10
x₂ = (40 - 20√6) / 10
6. Finally, simplify the solutions:
x₁ = 4 + 2√6
x₂ = 4 - 2√6
Therefore, the solutions to the equation 5/(x-8) - 5/x = 40/(x^2 - 8x) are x = 4 + 2√6 and x = 4 - 2√6.