to zero. is zero.

For if any number be multiplied by zero, the product

Either of these values of a must answer the conditions of the equation.

N. B. Though either value answers the conditions separately, they cannot be introduced together, for being different. their product cannot be x2.

Instead of m put its value, and the values of ≈ become

which are the values we had obtained above. (This demonstration is essentially that of M. Bourdon.)

Since the expression contains a radical quantity, that is, a quantity of which the root is to be found, in order to be able to find the value of it, we must be able to find the root either exactly or by approximation. Now there is one case in which

it is impossible to find the root. It is when q is negative and p2 In which case the expression q + 1o is ne

greater than

p2

4

4

gative; and it has been shown above, that it is impossible to find the root of a negative quantity. In all other cases the value of the equation may be found.

In all cases if q is positive, the first value will be positive, and answer directly to the conditions of the question proposed. P. be

For the radical (2 + 2) is necessarily greater than 2,

cause the root of alone is P ; therefore the expression

4

2

2(+2) is necessarily of the same sign as the

radical.

The second value is for the same reason essentially negative,

for both and (2 + 2) are negative. This value, though

2

4

it fulfils the conditions of the equation, does not answer the conditions of the question, from which the equation was derived; but it belongs to an analogous question, in which the x must be put in with the sign-instead of +; thus x-px=q, gives x = Ρ 士

2

which

a value, which differs from the

In order that it may be possible to find the root, q must be less than 2. When this is the case, the two values are real.

4

values are negative if p is positive in the equation; that is, if x2+px=-q, which gives

and both positive if p is negative in the equation, that is, x2.

q, which gives

When both values are negative, neither of them answers directly to the conditions of the question; but if x be put into the original equation instead of x, the new equation will show what alteration is to be made in the enunciation of the question; and the same values will be found for r as before, with the exception of the signs.

If in this equation q is greater than

P2, the quantity

4

becomes negative, and the extraction of the root

cannot be performed. The values are then said to be imagi

nary.

1. It is required to find two numbers whose sum is whose product is q.

Let x = one of the numbers,

This example presents the case above mentioned, in which

p and q are both negative.

The values are 9 and 6, both positive, and both answer the conditions of the question. And these are the two numbers required, for 96 15, 9 x 6 54. This ought to be = × = so, for a in the equation represents either of the numbers indifferently. Indeed whichsoever x be put for, p-x will represent the other; and px-x will be their product.

Again let p = 16 and q = 72.

16
x= ±
2

256

72) * = 8± ( — 8) *.

Here (— 8)a is an imaginary quantity, therefore both values

are imaginary.

In order to discover why we obtain this imaginary result, let us first find into what two parts a number must be divided, that the product of the two parts may be the greatest possible quantity. In the above example, p represents the sum of the two numbers or parts, let d represent their difference, then

d is greater than zero; but when d = 0, the expression becomes

P which is the second power of. Therefore the greatest

2

possible product is when the two parts are equal.

In the above example = 8, and p

p3 = 64.

This is the

2

4

greatest possible product that can be formed of two numbers whose sum is 16. It was therefore absurd to require the product to be 72; and the imaginary values of x arise from that absurdity.

2. It is required to find a number such, that if to its second power, 9 times itself be added, the sum will be equal to three times the number less 5.

This equation is in the form of a2 + p x = − q, which

px

The values are-1 and

-=

-3±2.

5, both negative. Consequently

neither value will answer the conditions of the question.

shows also that those conditions cannot be answered.

This

But if we change the sign of x in the equation, that is, put x instead of x, it becomes

in

This shows that the question should be expressed thus:

It is required to find a number, such, that if from 9 times itself, its second power be subtracted, the remainder will be equal to 3 times the number plus 5.