Note. In most cases the factors can be written down at once, as in Art. 79, without completing the square; and much labour is thereby saved.

121. Discussion of roots of a quadratic equation. In the preceding article we found that the quadratic equation ax + bx + c = 0 had two roots, namely

b2-4ac is positive or negative, it follows that the roots of ax2+ bx + c = 0 are real or imaginary according as b2 - 4ac is positive or negative.

The roots are clearly rational or irrational according as b2-4ac is or is not a perfect square. It should be remarked also that both roots are rational or both irrational, and that both roots are real or both imaginary.

If b2 - 4ac = 0, both roots reduce to

b

and are thus

2a, equal to one another. In this case we do not say that the equation has only one root, but that it has two equal

roots.

It is clear that the roots will be unequal unless b2-4ac0. Hence in order that the two roots of the equation ax2 + bx + c=0 may be equal, it is necessary and sufficient that b2 = 4ac.

When b2 = 4ac, the expression ax2 + bx + c is a perfect square in x, as we have already seen.

122. Special Forms. We will now consider some special forms of quadratic equations, in which one or more of the coefficients vanish.

II. If c = 0 and also b=0, the equation reduces to ax2= 0, both roots of which are zero.

III. If b = 0, the equation reduces to ax2 + c = 0, the

roots of which are ±

C
a

The roots are therefore equal

and opposite when b = 0, that is when the coefficient of x

is zero.

IV. If a, b and c are all zero, the equation is clearly satisfied for all values of x.

then we have, after multiplying by y',

cy2+by+a = 0.

Now from I. and II. one root of this quadratic in y is zero if a = 0, and both roots are zero if a = 0 and also b

1

But since x =

y

0.

x is infinity when y is zero. Thus one

root of ax2 + bx + c = O is infinite if a = are infinite if a = 0 and also b= 0.

Thus the quadratic equation

(a-a')x+(b-b')x+c-c'=0

0; also both roots

has one root infinite, if a=a'; it has two roots infinite, if a=a' and also bb'; and the equation is satisfied for all values of x, if a = a', b=b' and c=c'.

Again, the equation

a (x+b) (x+c)+ b (x+c) (x+a) = c(x+a) (x+b),

is a quadratic equation for all values of c except only when c=a+b, in which case the coefficient of x2 in the quadratic equation is zero. When c = a + b we may still however consider that the equation is a quadratic equation, but with one of its roots infinite.

Note. It is however to be remarked that since infinite roots are not often of practical importance, they are generally neglected unless specially required.

123.

Zero and infinite roots of any equation. The most general form of the equation of the nth degree is

ax" + bx"1+ ... + kx + 1 = 0..........(i).

If = 0, the equation may be written

x (αx11 + bx12 + ... + k) = 0,

one root of which is clearly zero.

Similarly two roots will be zero if l=0 and also k = 0 ; and so on, if more of the coefficients from the end vanish.

1

Put =; then we have, after multiplying by y",

У

a+by+......+ky”¬1 + ly" = 0.

From the above, one root of the equation in y will be zero when a = 0; and two roots will be zero if a= O and

also b=0. But when y = 0, x =

1

y

= ∞

Thus one root of (i) is infinite when a = 0, and two roots are infinite when a and b are both zero; and so on, if more of the coefficients from the beginning vanish.

124. Equations not integral. When an equation is not integral, the first step to be taken is to reduce it to an equivalent integral equation.

An equation will be reduced to an integral form by multiplying by any common multiple of the denominators of the fractions which it contains, but the legitimacy of this multiplication requires examination. For if we multiply both sides of an integral equation by an expression which contains the unknown quantity, the new equation will not only be satisfied by all the values of the unknown quantity which satisfy the original equation, but also by those values which make the expression by which we have multiplied vanish. Thus if each member of the equation A = B, be multiplied by P, the resulting equation PA = PB, or P (A – B) = 0, will have the same roots as the equation A-B=0 together with the roots of the equation P = 0.

When however an equation contains fractions in whose denominators the unknown quantity occurs, the equation may be multiplied by the lowest common multiple of the denominators without introducing any additional roots, for we cannot divide both sides of the resulting equation by any one of the factors of the L.C.M. without reintroducing fractions, which shews that there are no roots of the resulting equation which correspond to the factors of the expression by which we multiply.