Hence, to reduce radical equations, we deduce from these examples the following general RULE. Transpose the terms so that a radical part shall stand by itself; then involve each member of the equation to a power of the same name as the root; if the unknown quantity is still under the radical sign, transpose and involve as before; finally reduce as usual. 8 Reduce x-7=x+ 18-5. 168. Equations containing the unknown quantity involved to any power require Evolution in their reduction. 169. To reduce pure equations containing the unknown quantity involved to any power. Reduce the equation so as to have as one member the unknown quantity involved to any degree, and then extract that root of each member which is of the same name as the power of the unknown quantity. NOTE.It appears from the solution of Example 1 that every pure quadratic equation has two roots numerically the same, but with opposite signs. 14. Reduce 3x-1-5x-22x2+3x-1-Z. 15. Reduce (c+x)3 — 6 c2 x = (c — x)3 — 16 c3. 170. Equations containing radical quantities may require in their reduction both Involution and Evolution; and in this case the rule in Art. 167, as well as that in Art. 169, must be applied. Which rule is first to be applied depends upon whether the expression containing the unknown quantity is evolved or involved. 1. Reduce 17 — x3 — 2 = 12. 6. Reduce 2x2 + 8 x3 + 24 x2 + 32 x = x+2. 171. Equations which contain two or more unknown quantities may require for their reduction involution, or evolution, or both. In these equations the elimination is effected by the same principles as in simple equations. (Arts. 112-114.) |