113. If an equation involving one unknown quantity can be reduced to the form "N, the value of x can be found by simply extracting the nth root of both members, thus

114. Where it must be observed (Art. 79,) that when n is an even number, the value of x will be either plus or minus for all positive values of N, but for negative values of N the value of x will be impossible. When n is an odd number, the value of x will have the same sign that N has.

(115.) If the equation can be reduced to the form xm=N, then x can be found by raising both members to the mth power, thus:

x=Nm

Where x will be positive for all values of N, provided m is an even number, but when m is an odd number, then x will have the same sign as N.

(116.) Finally, when the equation can be reduced to

the form

We must first involve both members to the mth power, and then extract the nth root, or else we may first extract the nth root, and then involve to the mth power. (Art. 81.)

This equation, when cleared of fractions, by multiply

4. Given (ym-b)-a-d, to find the values of y.

Ans. y={(a—-d)2+b}m.

5. Given V-32-16-x, to find the value of x.

If we multiply the numerator and denominator of the left-hand member by the numerator, it will become

9. Given (2^—7)—3, to find the values of x.

10. Given (x-1)=-49, to find x.

Ans. x=2.

Ans. x 16.

is the most general form of a quadratic equation, where a= the coefficient of the first term,

b= the coefficient of the second term,
cthe term independent of x.

If we multiply the general quadratic equation (A) by 4a, it will become

4a2x2+4alx=4ac.

Adding b2 to both members of (1), it becomes

(1)

4a2x2+4ulx+b2=b2 +4ac. (2)

If we extract the square root of both members of (2),

Formula (B) may be regarded as a general solution

of all complete quadratic equations.

If we translate it into common language, we shall obtain a rule for solving complete quadratics.

Hence, to find the value of an unknown quantity, when given by a quadratic equation, we have this

RULE.

Having reduced the equation to the general form ax2+bx=c, we can find x, by taking the coefficient of the second term with its sign changed, plus or minus the square root of the square of the coefficient of the second term increased by four times the coefficient of the first term into the term independent of x; and the whole divided by twice the coefficient of the first term.

Four times the coefficient of the first term into the

term independent of x, is

4X4 X 36-576.