To solve the rational equation [tex]\(\frac{x-10}{x} = \frac{99}{x+10}\)[/tex], we need to find the values of [tex]\(x\)[/tex] that satisfy this equation. Here is the step-by-step solution:
1. Combine the fractions and simplify:
[tex]\[
\frac{x-10}{x} = \frac{99}{x+10}
\][/tex]
To eliminate the fractions, we cross-multiply:
[tex]\[
(x-10)(x+10) = 99x
\][/tex]
2. Expand both sides:
[tex]\[
x^2 - 100 = 99x
\][/tex]
3. Rearrange the equation:
[tex]\[
x^2 - 99x - 100 = 0
\][/tex]
4. Solve the quadratic equation [tex]\(x^2 - 99x - 100 = 0\)[/tex] using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -99\)[/tex], and [tex]\(c = -100\)[/tex]:
- Compute the discriminant:
[tex]\[
b^2 - 4ac = (-99)^2 - 4(1)(-100) = 9801 + 400 = 10201
\][/tex]
- Take the square root of the discriminant:
[tex]\[
\sqrt{10201} = 101
\][/tex]
- Apply the quadratic formula:
[tex]\[
x = \frac{-(-99) \pm 101}{2(1)} = \frac{99 \pm 101}{2}
\][/tex]
5. Calculate the solutions:
- For the positive root:
[tex]\[
x = \frac{99 + 101}{2} = \frac{200}{2} = 100
\][/tex]
- For the negative root:
[tex]\[
x = \frac{99 - 101}{2} = \frac{-2}{2} = -1
\][/tex]
Therefore, the solutions to the equation [tex]\(\frac{x-10}{x} = \frac{99}{x+10}\)[/tex] are:
[tex]\[
x = [-1, 100]
\][/tex]
