and, this done, found he had but 12s. remaining; what had he at first? Ans. 20s.

21. To divide the number 90 into 4 such parts, that, if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient, shall be all equal to each other. Ans. The parts are. 18, 22, 10, and 40, respectively.

22. The hour and minute hands of a clock are exactly together at 12 o'clock; when are they next together? Ans. 1 hour, 5 min,

23. When will the hour, minute, and second hands of a clock be all together next after 12 o'clock?

Ans. Only at 12 o'clock. 24. There is an island 73 miles in circumference, and 3 footmen all start together to travel the same way about it ; A goes 5 miles a day, B 8, and C 10; come together again?

when will they all Ans. 73 days.

25. If A can do a piece of work alone in 10 days, and A and B together in 7 days; in what time can B'do it alone? Ans. 23 days.

26. If three agents, A, B, and C, can produce the effects a, b, c, in the times e, f, g, respectively; in what time would they jointly produce the effect d?

27. If A and B together can perform a piece of work in 8 days, A and C together in 9 days, and B and C in 10 days; how many days will it take each person to perform the same work alone?

Vol. I.

Ans. A 14 days, B 1711, and C 234.

૨૧

QUADRATIC EQUATIONS.

A SIMPLE QUADRATIC EQUATION is that, which involves the square of the unknown quantity only.

AN AFFECTED QUADRATIC EQUATION is that, which involves the square of the unknown quantity, together with the product, that arises from multiplying it by some known quantity.

Thus, axb is a simple quadratic equation,

And ax+bx c is an affected quadratic equation.

The rule for a simple quadratic equation has been given already.

All affected quadratic equations fall under the three following forms.

1. x2+ax=b

2. x2-ax=b

3. x3-ax=-b..

The rule for finding the value of x, in each of these equations, is as follows.

RULE.

1. Transpose all the terms, that involve the unknown quantity, to one side of the equation, and the known terms to the other side, and let them be ranged according to their dimensions.

*The square root of any quantity may be either + or and therefore all quadratic equations admit of two solutions. Thus, the square root of +n is +n, or -n; for either +nx +n, or—nx—n is equal to +n2. So in the first form, where is found ✔ot. the root may be either +✔ote

x +

2

or -√6+ since either of them being multiplied by itself

a

will produce ¿+ a2. And this ambiguity is expressed by wri

2. When the square of the unknown quantity has any Coefficient prefixed to it, let all the rest of the terms be divided by that coefficient.

a2

ting the uncertain sign before 6+ —; thus x =

always be greater than✔, or its equal; and consequently

a2 a

The second value, viz. x——✔6+ ———, will always be

2

negative, because it is composed of two negative terms. There

fore, when x2+ax=b, we shall have x=+✔b+

the affirmative value of x, and x=-√✓ b+

2

composed of two affirmative tems. The second value, viz. x—

3. Add the square of half the coefficient of the second term to both sides of the equation, and that side, which involves the unknown quantity, will then be a complete

square.

a2

✔6+2+2 will always be negative; for since 6+22

is

+ is always a negative quantity. Therefore, when x2—ax

2

2

4

a

✦ first value, viz. x=+-+, is evidently affirmative, be

x=+√

4

ipg composed of two affirmative terms. The second value, viz.

+ is also affirmative; for since

both the values of

2

is greater than b.

For the

a

2'

4

quantity. Therefore, when a―ax-b, we shall have x=+

for the affirmative value of x, and

2

will always be an affirmative

4. Extract the square root from both sides of the equation, and the value of the unknown quantity will be determined, as required.

NOTE 1. The square root of one side of the equation is always equal to the unknown quantity, with half the coefficient of the second term subjoined to it.

NOTE 2. All equations, wherein there are two terms involving the unknown quantity, and the index of one is just double that of the other, are solved like quadratics by completing the square.

Thus, x*+ax2=b, or a"+axb, are the same as quadratics, and the value of the unknown quantity may be deter. mined accordingly.

EXAMPLES.

1. Given x2+4x=140; to find x.

First, x+4x+4=140+4=144 by completing the square;

But in this third form, if b be greater than

the proposed question will be impossible. For, since the square of any quantity (whether that quantity be affirmative or negative) is always affirmative, the square root of a negative quantity is

impossible, and cannot be assigned. But if b be greater than