What is the sum of all real solutions to the equation x−6/4=−2/x?
To find the sum of all real solutions to the equation x - 6/4 = -2/x, we need to solve the equation first.
Step 1: Multiply both sides of the equation by 4x to eliminate the fractions:
4x(x - 6/4) = -2
Step 2: Simplify the equation:
4x^2 - 6x = -2
Step 3: Move all terms to one side to get a quadratic equation:
4x^2 - 6x + 2 = 0
Step 4: Solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = -6, and c = 2.
Substituting these values into the quadratic formula, we get:
x = (-(-6) ± √((-6)^2 - 4(4)(2))) / (2(4))
= (6 ± √(36 - 32)) / 8
= (6 ± √4) / 8
= (6 ± 2) / 8
This gives us two potential solutions:
x1 = (6 + 2) / 8 = 8 / 8 = 1,
x2 = (6 - 2) / 8 = 4 / 8 = 1/2
So, the sum of all real solutions to the equation is 1 + 1/2 = 3/2.