𝕭𝖗𝖔 𝖙𝖍𝖎𝖘 𝖒𝖊𝖙𝖍𝖔𝖉 𝖎𝖘 𝖗𝖊𝖒𝖔𝖛𝖊𝖉 𝖋𝖗𝖔𝖒 𝖙𝖍𝖊 𝖘𝖞𝖑𝖑𝖆𝖇𝖚𝖘

To solve the quadratic equation (a b)x^2-2ax (a-b) = 0 using the completing the square method, we follow these steps:

Step 1: Rearrange the equation
Rearrange the equation in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. In this case, our equation is already in standard form, so we can proceed to the next step.

Step 2: Divide the equation by the coefficient of x^2
Divide the entire equation by the coefficient of x^2, which is (a b). This step helps simplify the equation and ensures that the coefficient of x^2 is 1. After dividing, the equation becomes:
x^2 - (2a/a b)x + (a-b)/(a b) = 0

Step 3: Complete the square for the x terms
To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation. In this case, the coefficient of x is -(2a/a b). So, we have:
x^2 - (2a/a b)x + (a-b)/(a b) = 0
x^2 - (2a/a b)x + (a-b)/(a b) + (2a/a b)^2 = (2a/a b)^2

Simplifying the left side:
x^2 - (2a/a b)x + (a-b)/(a b) + (4a^2/(a b)^2) = (4a^2/(a b)^2)

Step 4: Simplify and factor the perfect square trinomial
Combine the x terms and simplify the right side of the equation:
x^2 - (2a/a b)x + (a-b)/(a b) + 4a^2/(a b)^2 = 4a^2/(a b)^2
x^2 - (2a/a b)x + (a^2 - ab + 4a^2)/(a b)^2 = 4a^2/(a b)^2

Step 5: Factor the perfect square trinomial
Factor the perfect square trinomial on the left side of the equation:
(x - a/(a b))^2 = 4a^2/(a b)^2 - (a^2 - ab + 4a^2)/(a b)^2

Simplifying the right side:
(x - a/(a b))^2 = (4a^2 - (a^2 - ab + 4a^2))/(a b)^2
(x - a/(a b))^2 = (4a^2 - a^2 + ab - 4a^2)/(a b)^2
(x - a/(a b))^2 = (ab)/(a b)^2

Step 6: Take the square root of both sides
Take the square root of both sides of the equation:
x - a/(a b) = ±√(ab)/(a b)

Step 7: Solve for x
Now, isolate x by adding a/(a b) to both sides of the equation:
x