Explanation
To find the quadratic equation with the new roots, we use the method of transformation.
Step 1: Define the relationship
Let x be the root of the new equation. We are given the transformation:
x=α−1α+2
We need to express α in terms of x so we can substitute it into the original equation aα2+bα+c=0.
Step 2: Isolate α
x(α−1)=α+2
xα−x=α+2
xα−α=x+2
α(x−1)=x+2
α=x−1x+2
Step 3: Substitute into the original equation
Since α is a root of ax2+bx+c=0, we substitute the expression for α into the original equation:
a(x−1x+2)2+b(x−1x+2)+c=0
Multiply the entire equation by (x−1)2 to eliminate the denominator:
a(x+2)2+b(x+2)(x−1)+c(x−1)2=0
Step 4: Expand and Simplify
a(x2+4x+4)=ax2+4ax+4a
b(x2+x−2)=bx2+bx−2b
c(x2−2x+1)=cx2−2cx+c
Now, group the terms by the powers of x:
x2(a+b+c)+x(4a+b−2c)+(4a−2b+c)=0
Comparing this result with the given options, we can see it matches Option 4.
