THEORY OF QUADRATICS.

302. Having treated of the practical methods of solving quadratic equations, we will now proceed to consider certain general principles relating to quadratics.

303. Let us resume the general equation,

If we solve this equation, and represent one root by r and the other by r', we shall have

By adding these equations, and also multiplying them together, we

obtain

That is,

r+r=-2a,
rr' = —b.

(3)

(4)

1.- The sum of the two roots is equal to the coefficient of x taken with the contrary sign.

2. The product of the two roots is equal to the absolute term taken with the contrary sign.

304. From equations (3) and (4) in the last article, we have 2a (r+r), and b = -rr'.

=

Substituting these values in (A), and transposing the absolute term, we have

or by factoring,

x2—(r+r')x+rr' = 0 ;

(x—r)(x—r′) = 0.

Hence,

If all the terms of a quadratic equation be transposed to the first member, the result will consist of two binomial factors, formed by annexing the two roots with their opposite signs to the unknown quantity.

305. A Quadratic Expression is one which contains the first and second powers of some letter or quantity.

By the principle established in the preceding article, any quadratic expression whatever may be resolved into simple factors.

1. Let it be required to resolve the expression, x+12x-45, in

to simple factors.

Assume

x2+12x-450.

=

This equation readily gives

Hence,

x = 3, x=-15.

x2+12x-45(x-3) (x+15), Ans.

2. Separate 5x3-8x+3 into simple factors

We first separate the factor 5; thus,

We may now factor the quantity within the parenthesis, as in the last example; thus,

со

5'

8.x

x2

5

And the given quantity is factored as follows:
5x2-8x+3=5(x—1) (x—3), Ans.

1. Resolve x+2x-120 into simple factors.

X = − 1 + 11 = 10-12 Ans. (x-10) (x+12.)

2. Resolve x2-9x+14 into simple factors.

+

7082

4. Resolve x2-35x+300 into simple factors.

Ans. (x-2)(x-7)

Ans. (x+3)(x+5).

30

30

=

For 6. Resolve 15x+19x+6 into simple factors.

Ans. 15(x+3)(x+3).

7. Resolve cx-2ax+c'x-2ac' into simple factors.

306. The same principle also enables us to construct an equation whose roots shall be any given quantities. This is done by multiplying together the two binomial factors, which, according to the principle in question, the required equation must contain.

1. Find the equation whose roots shall be and —1.

1. Find the equation whose roots shall be 6 and -15.

Ans. x2+9x-90 = 0.

2. Find the equation whose roots shall be 3 and -15.

Ans. x2+12x-45 = 0.

3. Find the equation whose roots shall be 16 and 9.

Ans. x-25x+144 = 0.

4. Find the equation whose roots shall be 84 and -1. 4\x41

Ans. x2-83x-84 = 0.

5. Find the equation whose roots shall be and -.

223

6. Find the equation whose roots shall be 7 and

Ans. x

= 0.

56

2

Ans. 8x-6x+1=0.

7. Find the equation whose roots shall be and 1.

8. Find the equation whose roots shall be 2a and -C.

DISCUSSION OF THE FOUR FORMS.

307. In the general equation x2+2ux = b, the coefficient of x, as well as the absolute term, may be either positive or negative. Hence, to represent all the varieties, with respect to signs, we must employ the four forms, as follows:

We may now consider what conditions will render these roots real or imaginary, positive or negative, equal or unequal.

308. Real and imaginary roots.

In the first and second forms, the quantity a+b, under the rad ical, is positive, and the radical quantity is therefore real. But in the third and fourth forms, the quantity a-b, under the radical, will be negative when b is numerically greater than a'; in which case the radical quantity is imaginary. Hence,

1-In each of the first and second forms, both roots are always real.

2.- In each of the third and fourth forms, both roots are imagi nary when the absolute term is numerically greater than the square of one half the coefficient of x; otherwise they are real.

309. Positive and negative roots.

Since

a+ba' and a-ba', we have

Va2+b>a and Vaa—b < a

It follows, therefore, that the signs of the roots in the first and second forms will correspond to the signs of the radical; but the

signs of the roots in the third and fourth forms will correspond to the signs of the rational parts.

Hence,

1.- In each of the first and second forms, one root is positive and

the other negative.

2.-In the third form both roots are negative, and in the fourth form both roots are positive.

310. Equal and unequal roots.

It is obvious that in the first and second forms the two roots are always unequal; for in each of these forms, one root is numerically the sum of a rational and a radical part, and the other the difference of the same parts.

The same may be said of the third and fourth forms, if we except the case where a2 b ; in which case the roots are eqnal, and we have, for the third form,

x = a±0 — —a or —a,

and for the fourth form,

x = +a±0 = +a or +a.

Hence,

1.- In each of the first and second forms, the two roots are always unequal.

2.—In each of the third and fourth forms, the roots will be equal when the absolute term is numerically equal to the square of one half the coefficient of x; otherwise they will be unequal.

In the first and third forms, the negative root consists of the sum of the rational and radical parts; while in the second and fourth forms, the positive root consists of the sum of the two parts. Hence, if we exclude the case of equal roots,

3. In the first and third forms the negative root is numerically greater than the positive.

4-In the second and fourth forms, the positive root is numerically greater than the negative.

The principles which we have now established, respecting the roots of quadratic equations, are all that are of importance, either theoretically or practically.