Note. It must be remembered, in using the formula a(x-r1) (x - 2), that a represents the coefficient of x2 in the given expression; thus, in Ex. 2, we have a = 12.

284. Many expressions may be factored by the artifice of completing the square, in connection with Art. 111.

1. Factor a2 + a2b2 + b2.

By Art. 108, the expression will become a perfect square if the middle term is 2a2b2. Hence,

a* + a2b2 + b2= (a2 + 2a2b2 + b2) – a2b2

= (a2 + b2)2 – a2b2

=(a2+b2+ab) (a2 + b2 – ab), (Art. 111)

= (a2 + ab + b2) (a2 – ab + b2), Ans.

2. Factor 9 x2 - 39 x2 + 25.

9x - 39 x2 + 25 = (9x2 - 30x2+25) - 9x2

x* - x2 + 1 = (x2 + 2x2 + 1) - 3x2

=(x2 + 1)2 - (x3)2

=(x2+x√3+1) (x2 - x√3 + 1), Ans.

DISCUSSION OF THE GENERAL EQUATION.

285. The roots of the equation x2 + px = q are

1

-p+√p2 + 49 and r2 - p - √p2+4q.

2

2

We will now discuss these values for different values of p and q.

I. Suppose q positive.

Since p2 is essentially positive (Art. 192), the quantity under the radical sign is positive and greater than p2. Therefore the value of the radical is greater than p. Hence, r1 is positive and 2 is negative.

If p is positive, 2 is numerically greater than 1; that is, the negative root is numerically the greater.

If p is zero, the roots are numerically equal.

If p is negative, r1 is numerically greater than r2; that is,

the positive root is numerically the greater.

II. Suppose q = 0.

The quantity under the radical sign is now equal to p2, so

that the value of the radical is p.

If p is positive, r1 = 0, and r2 is negative.

If p is negative, 1 is positive, and r2 = 0.

III. Suppose q negative and 4 q numerically <p2.

The quantity under the radical sign is now positive and less than p2. Therefore the value of the radical is less than p.

If p is positive, both roots are negative.

If p is negative, both roots are positive.

IV. Suppose q negative and 4q numerically = p2.

The quantity under the radical sign is now equal to zero. Therefore the roots are equal; being negative if p is positive, and positive if p is negative.

V. Suppose q negative and 4q numerically >p2.

The quantity under the radical sign is now negative ; hence, by Art. 201, both roots are imaginary.

The roots are both rational or both irrational according as p2+4q is or is not a perfect square.

EXAMPLES.

286. 1. Determine by inspection the nature of the roots of the equation 2x2 - 5x – 18 = 0.

The equation may be written

9.

2

Since q is positive and p negative, the roots are one positive and the other negative; and the positive root is numerically the greater.

Determine by inspection the nature of the roots of the following:

XXVI. RATIO AND PROPORTION.

287. The Ratio of one quantity to another of the same kind is the quotient obtained by dividing the first quantity by the second.

Thus, the ratio of a to

bis a

which is also expressed a : b.

288. The first term of a ratio is called the antecedent, and the second term the consequent.

Thus, in the ratio a: b, a is the antecedent and b the consequent.

289. A Proportion is an equality of ratios.

Thus, if the ratio of a to b is equal to the ratio of c to d, they form a proportion, which may be written in either of the forms:

290. The first and fourth terms of a proportion are called the extremes; and the second and third terms the means.

Thus, in the proportion a:b=c:d, a and dare the extremes, and b and c the means.

291. In a proportion in which the means are equal, either mean is called a Mean Proportional between the first and last terms, and the last term is called a Third Proportional to the first and second terms.

A Fourth Proportional to three quantities is the fourth term of a proportion whose first three terms are the three quantities taken in their order.

Thus, in the proportion a:b=b:c, bis a mean proportional between a and e, and e is a third proportional to a and b. In the proportion a: b c:d, dis a fourth proportional to a, b, and c.