Roots of Equations.

(252.) A Root of an Equation is a value of the unknown quantity in the equation. It will presently be shown that an equation of the 2d degree has two roots, of the 3d degree three roots, and so on.

In the Simple Equation 3x=15, the value of the unknown quantity x is 5; then 5 is the root of the equation.

It is evident that in a Simple Equation there can be but one value of the unknown quantity that will satisfy the equation. An equation of the 1st degree has therefore but one root.

GENERAL PROPERTIES OF EQUATIONS.

1. Divisors of an Equation.

(253.) If a be a root of an Equation of any degree, containing but one unknown quantity, x; the equation-with all its terms transposed to one side-will be divisible by x-a.

Let a be a root of the Equation

then will the equation

x2+mx=s;

x2+mx-s=0 be divisible by x -a

For let r be the remainder, if any, after the quotient q has been obtained; then will

the dividend being equal to the remainder added to the product of the divisor and quotient.

But a being a value of x, (252), we have x-a=0; then the (x-a) x q is 0, (43); and consequently r=0; that is, the division will leave no remainder.

The preceding demonstration is applicable to an equation of the third, fourth, or any higher degree.

(254.) Conversely, If an Equation of any degree, containing but one unknown quantity, x,—with all its terms transposed to one side→ be divisible by x-a; then a will be a root of the equation.

This is evident from considering that the divisibility of the Equation, as shown above, depends on the condition that x-a=0, or that a is a value of r.

2. Number of Roots of an Equation.

(255.) Every Equation containing but one unknown quantity, has just as many roots as there are units in the exponent of the highest power of the unknown quantity in the equation.

Let a represent a root of the cubic Equation

x3+mx2+nx=s.

Transposing s to the first side of the equation, we have x3+mx2+nx—s=0.

Dividing this equation by x-a, (253), we shall obtain an equation of the second degree, which may be represented by

x2+px-q=0.

--b.

Let b be a root of this equation. Dividing the equation by x-we shall obtain an equation of the first degree, represented by

x-u=0.

The binomials x-a, x-b, and x-u, which represent the two di visors and the last quotient, are the factors of the dividend.

x3+mx2+nx—s.

The original equation is thus resolved, representatively, into (x—a)(x—b)(x—u)=0.

Since this equation is divisible by each of these three binomial fac tors, it follows that a, b, and u are three roots of the equation, (254).

And since the given equation-being of the 3d degree—cannot be resolved into more than three binomial factors, each containing the first power of x, it cannot have more than three roots.

The same method of demonstration will show the proposition to be true for an Equation of any other degree.

The several roots of an Equation are not necessarily unequal, though such will usually be found to be the case. The preceding Proposition shows that an Equation may be resolved into binomial factors, each containing a root of the equation. Two or more of these roots may be equal to each other.

In the Equation x2 -5x+6=(x−2)(x−3)=0,

the two roots are 2 and 3.

In the Equation x3-7x2+16x-12=(x-2)(x-2)(x-3)=0
the three roots are 2, 2, and 3.

人

SOLUTION OF PURE EQUATIONS OF THE SECOND AND
HIGHER DEGREES.

The Equations belonging to this class are those which, in their simplest forms, contain but one power of the unknown quantity

RULE XXII.

(256.) For the Solution of a Pure Equation.

1. Reduce the Equation to the form

x"=s; in which x" must be positive.

2. Extract that root of both sides of the equation which corresponds to the power of the unknown quantity

Dividing both sides of this equation by the coefficient —7,

The two values of x are thus found to be 4 and

of which will satisfy the given Equation.

It is evident that, in a Pure Equation of the second degree, the

two values of the unknown quantity will always be equal, with con

trary signs.

The unknown quantity may enter an Equation in a surd expression, which it will be necessary to rationalize in the solution of the equation.

Thus in the equation √x+1+x=a, it would be necessary to rationalize x, that is, to clear it of the radical sign before the value of a could be determined.

The following observations will assist the student in the

RATIONALIZATION OF SURD QUANTITIES IN AN EQUATION. 171

Rationalization of Surd Quantities in an Equation.

(257.) A Surd quantity in an Equation will be rationalized by transposing all the other terms to the other side of the equation, and raising both sides to the power corresponding to the indicated root.

To rationalize x in the Equation

√x+1-a=b.

By transposition,

√x+1=a+b
x+1=a2+2ab+b2.

Squaring both sides,

(258.) Two Surds in an Equation may be rationalized by successive involutions,—in the first of which it will generally be expedient to make one of the Surds stand alone on one side of the equation.

(259.) When an Equation contains a Fraction whose terms are both irrational, it will sometimes be expedient to rationalize its denominator before clearing the equation of the fraction.

Multiplying both terms of the Fraction by 1-√x, (243...2),

Clearing this equation of its Fraction, and transposing, we have x=a-ax+x.

The Surd in this equation will be rationalized by squaring both

sides, as in the preceding examples.

By the preceding methods we may generally rationalize one or more Surds containing the unknown quantity in an Equation. Other expedients, however, such as the extraction of roots in possible cases, in the course of the operation, will sometimes be requisite; but these must be left to the care and skill of the computer.

5. Find the value of x in the equation

√x-32=√/x-√32.

6. Find the value of x in the equation

3+√x+4×√x−4=10.

7. Find the value of x in the equation

a+√x−3 × √x+3=4a.

Ans. x=50.

Ans. x=±√65.

Ans. x=3a2+1.

Ans. x=13