Answer: x = 20 and x = 4.
Step-by-step explanation: 1. Multiply every term by x to clear the denominator:
x(x + 80/x) = 24x
x^2 + 80 = 24x
2. Rearrange the equation to set it equal to zero, to solve for x using the quadratic formula:
x^2 - 24x + 80 = 0
3. Use the quadratic formula, where a = 1, b = -24, and c = 80:
x = [ -(-24) ± √((-24)^2 - 4(1)(80)) ] / 2(1)
4. Simplify and solve for x:
x = [ 24 ± √(576 - 320) ] / 2
x = [ 24 ± √256 ] / 2
x = [ 24 ± 16 ] / 2
5. There are two possible solutions:
x1 = (24 + 16) / 2 = 40/2 = 20
x2 = (24 - 16) / 2 = 8/2 = 4
Therefore, the solutions to the rational equation x + 80/x = 24 are x = 20 and x = 4.