If α and β are the roots of 2x²-7x-6=0 find the equation where whose roots are (2α²+3) and (2β²+3)
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Answer:
If α and β are the roots of equation aX^2+bX+c=0 then how do you express the roots of (X+2) +1/x=b^2/ac in terms of α and β?
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Noodling around, here is what I came up with:
Ax^2 + Bx + C = 0 has roots a,b. Using Viete’s formulas, B/A = a + b and C/A = ab.
So c = Aab, and B/C = B/(Aab) = B/A * 1/ab = (a + b)/ab.
considering x + 2 + 1/x = b^2/ac, note b^2/ac = b/a * b/c = (a + b)^2/ab. Call this k.
Multiply by x: x^2 + 2x + 1 = kx. Rearrange: x^2 + (2-k)x + 1 = 0.
The roots of this are (k-2 +/- sqrt(4 - 4k + k^2 -4))2 = (k-2 +/- sqrt(k^2 - 4k))/2, where k is the combination of roots noted above.
Caveats: x cannot be zero due to the form of the equation, and multiplying by x is not an equivalence operation, and could introduce extraneous roots, so the solution gives only potential roots. They must be verified to be accepted.