3. Multiply a+vx by 3b + √2x.

Ans. 3ab3bvx + a√2x + √2x2.

or,

3ab+3bvx+a√2x+x√2.

4. Multiply va + va +x by 2√3a +3x.

Ans. 2√3a2+3ax+2(a + x)√3.

8. Divide 3ab + 3bvx + av2x+v2x2 by a + vx.

11. Involve √3a+v2a to the 2d power.

Ans. 3a6ax + 2x.

12. Involve 2v-3v to the 2d power.

Ans. 4x-12x+9x=1

13. Extract the square root of 9a +36√3ax+108x.

14. Extract the cube root of a3 +3a2√x +3a√x2+x.

16. Extract the square root of a2+4avax +6ax + 4x vax Ans. a +2vax+x.

EQUATIONS INVOLVING RADICAL QUANTITIES.

(70.) An equation may contain a root of the unknown quantity instead of that quantity itself; and in such a case, the rules given for the reduction of simple equations are not sufficient. The equation ve+a3b-c2 is one of this kind. For the resolution of equations, containing a root of the unknown quantity, we give the following

RULE.

1. Transpose the terms of the equation, so that the radical may stand alone on one side of the sign of equality.

2. Then involve both members of the equation to the power denoted by the index of the radical: the resulting equation will contain no radical, and may be reduced by the rules already given.

This rule depends upon the principle, that if two quantities are equal, any powers of these quantities will also be equal; and hence the equation will not be destroyed by involving both members to the same power.

2. Given v4+x=4—√x, to find x.

Squaring both members of this equation, we have,

Clearing this equation of fractions, we have,

√2ax + x2+2a + x = 4a;

and transposing, v2ax+x=2a-x.

Squaring both members,

4. Given v4a2 + x2 = √4b2 + x1, to find x. Squaring both members we have,

4a2 + x2 = v4b2 +xa.

Squaring both members a second time,

a, to find x.

5. Given x√a2+xvb2 + x2

Transposing, x+ a = √a2 + xvb2 + x2
Squaring both members,

x2+2ax+a2 = a2 + xvb2 + x2 ;

7. Given√5 +12= 17, to find x.

3

8. Given v12+x=2+vx, to find x.

X.

x=15.

= 4.

EQUATIONS OF THE SECOND DEGREE.

(71.) An equation of the form a2=a, or ax2 + bx = c, containing the unknown quantity in the second power, is termed an equation of the second degree. An equation which contains x in no other power besides the second, as

x2

+ x2 = ab, is called an equation of two terms, for it can 2 always be reduced to such a form, that the square of the unknown quantity may constitute one member of the equation, and the known quantities the other.

x2

2

+2ab may be