After having arranged the polynomial with reference to a, extract the square root of 25a1, this gives 5a2, which is placed at the right of the polynomial; then divide the second term, -30a3b, by the double of 5a2, or 10a2; the quotient is -3ab, and is placed at the right of 5a2. Hence, the first two terms of the root are 5a2-3ab. Squaring this binomial, it becomes 25a1-30a3b+9a2b2, which, subtracted from the proposed polynomial, gives a remainder, of which the first term is 40a2b2. Dividing this first term by 10a2, (the double of 5a2), the quotient is +462; this is the third. term of the root, and is written on the right of the first two terms. By forming the double product of 5a2-3ab by 4b2, and at the same time squaring 462, we find the polynomial 40a2b2-24ab3+16b4, which, subtracted from the first remainder, gives 0. Therefore 5a2-3ab+462 is the required

root.

2. Find the square root of a1+4a3x+6a2x2+4ax3+x1.

QUEST.-114. Give the rule for extracting the square root of a polynomial? What is the first step? What the second? What the third ? What the fourth?

6. What is the square root of

x1-4ax3+4a2x2-4x2+8ax +4.

Ans. x2-2ax-2.

7. What is the square root of

9x2-12x+6xy+y2-4y+4.

8. What is the square root of y1—Qy2x2+2x2—2y2+1

Ans. 3x+y-2.

Ans. y2-x-1.

9. What is the square root of

9a4b4-30a3b3+25a2b2?

Ans. 3a2b2—5ab.

10. Find the square root of

25a4b2-40a3b2c+76a2b2c2-48ab2c3+36b2c4-30a1bc

+24a3bc2-36a2bc3+9ac2.

Ans. 5a2b-3a2c-4abc +6bc2.

115. We will conclude this subject with the following remarks.

1st. A binomial can never be a perfect square, since we know that the square of the most simple polynomial, viz: a binomial, contains three distinct parts, which cannot experience any reduction amongst themselves. Thus, the expression a2+b2 is not a perfect square; it wants the term +2ab in order that it should be the square of a±b.

2nd. In order that a trinomial, when arranged, may be a perfect square, its two extreme terms must be squares, and the middle term must be the double product of the square roots of the two others. Therefore, to obtain the square root of a trinomial when it is a perfect square; Extract the roots of the two extreme terms, and give these roots the same or contrary signs, according as the middle term is positive or

negative. To verify it, see if the double product of the two roots gives the middle term of the trinomial.

since

Thus,

9a6-48a4b2+64a2b4 is a perfect square,

√9a3a3, and √64a2b4-8ab2,

and also 2 × 3a3 × — 8ab2 – -48a4b2- the middle term.

But 4a2+14ab+962 is not a perfect square: for although 4a2 and +962 are the squares of 2a and 3b, yet 2 × 2a × 3b is not equal to 14ab.

3rd. In the series of operations required in a general example, when the first term of one of the remainders is not exactly divisible by twice the first term of the root, we may conclude that the proposed polynomial is not a perfect square. This is an evident consequence of the course of reasoning, by which we have arrived at the general rule for extracting the square root.

4th. When the polynomial is not a perfect square, it may be simplified (See Art. 104.)

Take, for example, the expression √a3b+4a2b2+4ub3. The quantity under the radical is not a perfect square; but it can be put under the form ab(a2+4ab+4b2). Now, the factor between the parenthesis is evidently the square of a+2b, whence we may conclude that,

√a3b+4a2b2+4ab3=(a+2b) √ab.

2. Reduce 2a2b-4ab2+2b3 to its simple form.

Ans. (a-b)√26.

QUEST.-115. Can a binomial ever be a perfect power? Why not? When is a trinomial a perfect power? When, in extracting the square root we find that the first term of the remainder is divisible by twice the root, is the polynomial a perfect power or not?

CHAPTER VI.

Equations of the Second Degree.

116. An equation of the second degree is an equation involving the second power of the unknown quantity, or the product of two unknown quantities. Thus,

x2=a, ax2+bx=c, and xy=d2,

are equations of the second degree.

117. Equations of the second degree are of two kinds, viz: equations involving two terms, which are called incomplete equations; and equations involving three terms, which are called complete equations. Thus,

x2a and ax2=b,

=

are incomplete equations; and

x2+2ax=b, and ax2+bx=d,

are complete equations.

QUEST.-116. What is an equation of the second degree?-117. How many kinds are there? What is an incomplete equation? What is a complete equation?

118. When we speak of an equation involving two terms, and of an equation involving three terms, we understand that the equation has been reduced to its simplest form.

although in its present form there are four terms, yet it may be reduced to an equation containing but two. For, by adding 3x2 to 4x2 and transposing-4, we have

Also, if we have

7x2-10.

3x2+5x+7x+5=9,

we get by reducing

3x2+12x=4,

an equation containing but three terms.

Again, if we take the equation

QUEST. 118. When you speak of an equation involving two terms, do you speak of the equation after it has been reduced, or before? When you speak of an equation of three terms, is it the reduced equation to which you refer? To what forms, then, may every equation of the second degree be reduced?