CHAPTER IX. EQUATIONS OF THE SECOND DEGREE. I, EQUATIONS CONTAINING BUT ONE UNKNOWN QUANTITY Reduction to Particular Form. 140. Any equation of the second degree, containing but one unknown quantity, can always be reduced to the form of 2? + 2px = 4. clearing of fractions, and performing indicated operations, we have, 6x2 18x + 60 = 72 8x? + 16x2 + 16x + 4; transposing the unknown terms to the first member, and the known terms to the second member, we have, dividing by the coefficient of a?, that is, by – 2, which is of the required form, 2p, in this case, being equal to 17, and q being equal to – 8. All other equations of the same kind may be treated in the same manner; hence, we have the following rule for reducing equations of the second degree, containing but one unknown quantity, to the form 22 + 2px - q: RULE. 1. Clear the equation of fractions, and perform all the indicated operations. II. Transpose all the unknown terms to the first member, and all the known terms to the second member. III. Reduce all all the terms containing the square of the unknown quantity to a single term, one factor of which is the square of the unknown quantity ; reduce, also, all the terms containing the first power of the unknown quantity to a single term. IV. Divide both members of the resulting equation by the coefficient of the square of the unknown quantity. The resulting equation, is called the reduced equation. Solution of the Reduced Equations. 141. The solution of the reduced equation, consists in finding such values of the unknown quantity as will satisfy it, that is, when substituted for the unknown quantity, will make the two members equal. Every such value is called a root. Two cases may arise : first, it may happen that 2p, or the coefficient of the first power of the unknown quantity, is equal to 0; in this case, the equation is said to be incomplete : secondly, it may happen that the coefficient of the first power of the unknown quantity is not equal to 0; in this case the equation is said to be complete. Incomplete equations, when reduced, have but two terms : one containing the square of the unknown quantity; the other, a known term. Complete equations, when reduced, have three terms, viz.: a term containing the square of the unknown quantity, a term containing the first power of the hown quantity, and a known term. First Case. Incomplete Equations. 142. In this case, the reduced equation takes the form, 2 = a; extracting the square root of both members, we have, hence, we have the following rule for solving incomplete equations : RULE. Reduce the equation to the form, x? = 9, and extract the square root of both members. There will be two roots numerically equal, but having contrary signs. Denoting the first root by x', and the second by a'', we have, = 117 2. 3.x2 4= 28 + x2. Ans. c = + 4, ad" = 4. 322 + 5 x2 + 29 3. - 5x2. 8 3 Ans. x' = + 5, 2'' = – 5. 4. x2 + ab = 5x?. Ans. x = + Vab, x = - Vab. 5. (x + a)? = 2ax + b. Ans. x' = + vb – a?, " = -Vo - a?. x + 7 XC ing ny 6. x2 78 x2 + 7x x? – 73 Ans. Q = + 9, 7. xva + 2 = b + 2%. Ans. x' =+ va - 26 Second Case. Complete Equations. 143. The reduced form of the complete equation is, 2 + 2px = 1; adding pa to both members, (axiom 1°), we have, 22 + 2px + p2 = q + p2; extracting the square root of both members, (axiom 5o), we have, X + p= +Va + p?; transposing p to the second member, we have, x = -PEVI + p2; hence, there are two roots, one corresponding to the plus sign of the radical, and the other to the minus sign; denoting these roots by x' and a", we have, a' = -p++ p, and 2" = -p-19 + p?; hence, we have the following rule for solving complete equations of the second degree: RULE. I. Reduce the equation to the form, 2+2px = 9, by the rule. II. The first root is equal to half the coefficient of the second term, taken with a contrary sign, plus the square root of the second member increased by the square of half the coefficient of the second term. |