Since 2 —3 is a root, we know that 2 +3 is also a root, and corresponding to this pair of roots we have the quadratic factor x^2 -4x +1.
Also
624 — 13.28 — 3522-243 =(@?-42+41)(6224 1lx+3); hence the other roots are obtained from 6224 112+4+3=0, or (82%+1)(2x%+3)=0; thus the roots are —4, — 3, 2+,/3, 2— /3.
Ex. 2. Form the equation of the fourth degree with rational coefficients, one of whose roots is 2 +—3.
Here we must have 2 + —3, 2——3 as one pair of roots, and — 2 +—3, —2 - -3 as another pair.
Corresponding to the first pair we have the quadratic factor x^2 —22x+5, and corresponding to the second pair we have the quadratic factor x^2 + 22x+5.
Thus the required equation is
(a? + 2./2% 4 5)(—-—2V2a+ 5)=9, or (22 + 5)? — 82? =0, or ot +222 4+ 25=0.
EXAMPLES XLVIII. c.
Solve the equations :
1. 32¢ — 1023 + 422 x — 6 =0, one root being ee Ss
2. xt — 3622 + 72” — 36 = 0, one root being 3 — 1/3. 3. af +4224 502 +22 —2=0, one root being — 14+V—1. 4, xt4+493 + 602+4a”+45=0, one root being V— 1.
TRANSFORMATION OF EQUATIONS.
579. The discussion of an equation is sometimes simplified by transforming it into another equation whose roots bear some assigned relation to those of the one proposed. Such transformations are especially useful in the solution of cubic equations.
