Find 1/(a - 1) + 1/(b - 1) where a and b are the roots of the quadratic equation 2x^2-7x+2 = x^2-11x+1.
+2668 Simplify the equation to x2+4x+1=0
Note that1(a−1)+1(b−1)=(b−1)(a−1)(b−1)+(a−1)(a−1)(b−1)=a+b−2ab−a−b+1=(a+b)−2ab−(a+b)+1
Now, recall that a+b=−ba=−4 and ab=ca=1.
Substituting this in gives us −4−21+4+1=−66=−1
+289 Here is a different solution:
Root transformation!
2x^2 - 7x + 2 = x^2 - 11x + 1
x^2 + 4x - 1 = 0
If a, b are the solutions. Then a-1, b-1 are the solutions to the equation below:
(x+1)^2 + 4(x+1) - 1 = 0
x^2 + 6x + 4 = 0
The reciprocal of the roots of this equation would be reversed:
4x^2 + 6x + 1 = 0.
The sum of the roots is -6 by vietas.
