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

transposing the unknown terms to the first member, and the known terms to the second member, we have,

6x2+8x2 16x2 18x 16x 72+ 4 — 60;

factoring and reducing, we have,

2x2

· 34x = 16;

dividing by the coefficient of x2, that is, by — 2,

which is of the required form, 2p, in this case, being equal to 17, and 7 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

I. 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.

the

III. Reduce all the terms containing 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 unknown quantity, and a known term.

142. In this case, the reduced equation takes the form,

extracting the square root of both members, we have,

x = ± √q;

hence, we have the following rule for solving incomplete equations:

RULE.

Reduce the equation to the form, x2 = q, 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 x', we have,

Ans. x' = +Vab, x" = — √ab.

Second Case. Complete Equations.

143. The reduced form of the complete equation is, x2 + 2px = 9;

adding p2 to both members, (axiom 1°), we have,

x2 + 2px + p2 = q + p2;

extracting the square root of both members, (axiom 5°), we have,

x + p = ± √ q + p2 ;

transposing p to the second member, we have,

x = − p ± √ q + 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 x", we have,

x' = − p + √ q + p2, and x" = − p − √ q + p2 ;

hence, we have the following rule for solving complete equations of the second degree:

RULE.

I. Reduce the equation to the form, x2+2px = q, 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.