Px^2-(2p+q)x+(p+q)=0

solve by completing of square method

To solve the quadratic equation using the completing the square method, you need to follow these steps:

Step 1: Rewrite the equation
Rearrange the equation in the standard quadratic form: ax^2 + bx + c = 0.

In this case, the given equation is:
Px^2 - (2p+q)x + (p+q) = 0

Step 2: Divide by the coefficient of x^2
Divide the entire equation by the coefficient of x^2 to make the leading coefficient equal to 1.

Dividing the equation by P, we get:
x^2 - (2p+q)/P x + (p+q)/P = 0

Step 3: Move the constant term to the other side
Move the constant term (the term with no 'x') to the right side of the equation so that you have only the terms with 'x' on the left side.

The equation becomes:
x^2 - (2p+q)/P x = -(p+q)/P

Step 4: Complete the square
To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation.

To find the term we need to add and subtract on the left side to complete the square, we need to take half of the coefficient of x, square it:
(1/2)(2p+q)/P = (2p+q)/(2P)

Now add and subtract (2p+q)/(2P) on the left side:
x^2 - (2p+q)/P x + (2p+q)/(2P) - (2p+q)/(2P) = -(p+q)/P

Step 5: Factor the left side
Factor the left side of the equation as a perfect square trinomial.

(x - (2p+q)/(2P))^2 = -(p+q)/P - (2p+q)/(2P)

Simplify the right side:
(x - (2p+q)/(2P))^2 = (-2p-2q)/(2P)

Step 6: Solve for x
To solve for x, take the square root of both sides of the equation.

x - (2p+q)/(2P) = ± √[(-2p-2q)/(2P)]

Step 7: Simplify
Simplify the expression on the right side of the equation.

x - (2p+q)/(2P) = ± √[-(2p+2q)/(2P)] (Take out the common factor of 2)
= ± √[-(p+q)/P]

Step 8: Solve for x
Finally, solve for x by adding (2p+q)/(2P) to both sides of the equation.

x = (2p+q)/(2P) ± √[-(p+q)/P]

This is the solution to the quadratic equation Px^2 - (2p+q)x + (p+q) = 0 using the completing the square method.