From these illustrations, we derive the following precepts for the solution of radical equations : 1.-1t is sometimes advantageous to rationalize the denominator of a fractional term, before transposition or involution. 2.-An equation should be simplified as much as possible before involution; and care should be taken so to dispose the terms in the two members as to secure the simplest results after involution. EXAMPLES FOR PRACTICE. Find the values of the unknown quantity in each of the following equations : 1. Væ+7+vx=7. Ans. x = 9. 2. 2+3 =Vz4z+59. Ans. x = 5. 8ECTION V. QUADRATIC EQUATIONS. 283. A Quadratic Equation is an equation of the second degree, or one which contains the second power of the unknown quantity, and no higher power; as 3z* = 48, and ax—26x = c. That term of the equation which does not contain the unknown quantity, is called the absolute term. 284. Quadratic equations are divided into two classes-Pure Quadratics, and Affected Quadratics. PURE QUADRATICS. 285. A Pure Quadratic Equation is one which contains the second power only, of the unknown quantity; as 3x*—7 = 20. NOTE.-A pure equation, in general, is one which contains but a single power of the unknown quantity. 286. It is evident that by the rule for solving simple equations, every pure quadratic may be reduced to the form of 2 = , in which a may be any quantity, real or imaginary, positive or negative. Extracting the square root of both members of this equation, we have, x = ty a or-ya Hence, Every pure quadratic equation has two roots, equal in numerical value, but of opposite signs. OPERATION. 32; 2-4 20% 4 2 clearing of fractions, 20-8-32+72=63-384; collecting terms, 7* = 448; dividing by 7, x = 64; by evolution, We have therefore, for the solution of pure quadratics, the following RULE. Reduce the equation to the form of x'=a, and then take the square root of both members. +8, Ans. EXAMPLES FOR PRACTICE. Find the values of in each of the following equations : 1. 3x*—16 = 2*12. Ans. x=+3. 2. 23* _54 = 126-3x*. Ans. x = +6. 3. 70* +8 = 57 +32° +15. Ans. x = +4. -2d 4. bx+2d = 2cc ta. Ans. 2 = + 642c 5. axé +1 = (a—2) (a+x). Ans. x = +Va-1. 3+4 4 10 6. + Ans. x=+8. x+4 39 2+2 X-2 13 7. + Ans. x= + 10. +2 6 X (245)(x+5) =*-4. Ans. x= +4.54924+. x+4 AFFECTED QUADRATICS. 287. An Affected Quadratic Equation is one which contains both the first and the second powers of the unknown quantity; as 2x -3x = 12. Notes. 1. The two classes of quadratics, pure and affected, are sometimes called, respectively, incomplete and complete equations of the second degree. 2. A complete equation, in general, is one which contains every power of the unknown quantity, from the first to the highest inclusive. Thus a complete equation of the third degree must contain the first, second, and third powers of the unknown quantity. 288. Every affected quadratic equation may be reduced to the general form, ac* +2ax = b, in which 2a and b are positive or negative, integral or fractional. For, to effect this, we have only to bring all the terms containing the unknown quantity into the first member, and all the known terms into the second member, and divide the result by the coefficient of x'. 289. To solve a quadratic, suppose it first reduced to the form, **+2ax = b. To both members add a', square one half the coefficient of x; thus **+2axta’ = a*+b. or the of |