F.4 Quadratic Equation

Let the roots of 2x^2 - 3x - 2 = 0 be A , B .

then the roots of the require equation are :

2A+1 and 2B+1 .

sum of the roots = (2A+1)+(2B+1) = 2(A+B)+2

product of the roots = (2A+1)(2B+1) = 4AB + 2B + 2A + 1= 4AB+2(A+B)+1

Since A+B = 3/2
AB = -2/2 = - 1

so sum of the roots = 2(3/2)+2 = 5
product of the roots = 4(-1) + 2(3/2) + 1 = 0

The required equations becomes x^2 - 5x + 0 = 0

x^2 - 5x = 0

2x - 3x -2 = 0 is not a quadratic equation
You mean 2x^2 - 3x -2 = 0 ? Assume this is your equation:

For the equation 2x^2 - 3x -2 = 0,
Let the roots be α and β.
Sum of roots = α+β = (-3)/2 = 3/2
Product of roots = αβ = (-2)/2 = -1

For the new equation,
Sum of roots = (2α+1)+(2β+1) = 2(α+β)+2 = 2(3/2)+2 = 5
Product of roots = (2α+1)(2β+1) = 4αβ+2α+2β+1 = 4(-1)+2(α+β)+1 = -4+2(3/2)+1= 0

So, the new required qudratic equation is:
x^2 - (sum of roots)x + (product of roots) = 0
x^2 - 5x + 0 = 0
x^2 - 5x = 0

2009-11-01 03:39:29 補充：
α = A
β = B