SECTION 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 3x2 = 48, and ax2 2bx = 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 3x2720.

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

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

Hence,

x = + √a or - √a.

Every pure quadratic equation has two roots, equal in numerical value, but of opposite signs.

clearing of fractions, 2x2-8- 3x2 + 72 = 6x2 — 384 ;

collecting terms,

dividing by 7,

by evolution,

7x2 = 448;

x2 = 64;

x= ± 8, Ans.

We have, therefore, for the solution of pure quadratics, the following

RULE.-Reduce the equation to the form of x2= a, and then take the square root of both members.

Find the values of x in each of the following equations:

287. An Affected Quadratic Equation is one which contains both the first and the second powers of the unknown quantity; as 2x2 — 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,

x2+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 x2.

289. To solve a quadratic, suppose it first reduced to the form,

x2 + 2ax = b.

To both members add a2, or the square of one half the coefficient of x; thus

x2 + 2ax + a2 = a2 + b.

The first member is now a complete square. Taking the square root of both members, we have

x + a = ± √ a2 + b ;

whence by transposition,

x== a± √ a2 + b.

Thus the equation has two roots, which are unequal in all cases except when a2 + b = 0; in this case we shall have

x = a ±0,

and the equation is said to have two equal roots. Thus take the equation,

Hence, for the solution of an affected quadratic equation we have the following

RULE.-I. Reduce the given equation to the form of x2+2ax = b. II. Add to both sides of the equation the square of one-half the coefficient of x, and the first member will be a complete square.

III. Extract the square root of both members, and solve the resulting equation.

290. When the equation has been reduced to the form of

x2 + 2ax = b,

its roots may be obtained directly by the following obvious rule: Write one-half the coefficient of x with its contrary sign, plus or minus the square root of the second member increased by the square of one-half this coefficient.

Find the values of the unknown quantity in each of the follow