QUADRATIC EQUATIONS.

IF, after all the unknown quantities, except one, are exterminated from an equation, both that unknown quantity and its square are found in it, the equation is called a Quadratic.

TO RESOLVE A QUADRATIC EQUATION.

Having cleared the equation, and brought the terms involving the unknown quantity to one side of it by themselves, divide by the coefficient of the square of the unknown quantity, if it have one; then add to both sides the square of half the coefficient of the unknown quantity, which will complete the square of the side containing the unknown quantity; after which extract the square root of both sides, and the equation will be reduced to a simple one, which may be resolved as before.

NOTE 1.-Since the square root of x2 2ax+a2 is either ax or x- -a, the root of the known side of the equation must have both the signs and before it. Sometimes both these give proper solutions, and at other times only one of them. NOTE 2. The root of the side involving the unknown quantity consists of that quantity, and of its coefficient with its sign.*

1. Let the equation be 3x2 + 12x By dividing by 3, x2+4x

.

Add the square of 2,

And taking the root,

And transposing,

=

= 96

32

x=±6—2=+4 or—8.

Here the positive value of the root only is proper.

2. Let the equation be 2x2 8x

x2 4x

Completing the square, x2-4x+4=49

Dividing by 2,

= 90

= 45

Here also the root 7 is greater than the coefficient of x; therefore the positive value only is proper.

* Quadratic equations assume one of these three forms, viz. x2+ax=+b; x2-ax=+b; or x2-ax=-b; and when they are resolved by the rule, the value of x assumes one of these forms, -a±√a2+4b +a±√ a2+4b +a±√a2. -4b

x=

2

; x =

; or x=

2

If a positive answer is required, the sign of the radical in the first two forms must be +, but in the third it may be either + or - There is, however, a limitation in this case, for 46 must not be greater than a2, otherwise the quantity below the radical sign would be negative, and its root impossible. This happens when the absolute term b is greater than 4a2, the square of the coefficient of x.

Here both the roots are proper. But it is to be remarked,

225
49

the known side would

that if 54 had been greater than have been negative, and its root impossible; in which case x would have had no value in numbers..

NOTE.-To avoid fractions, instead of dividing by the coefficient of x2, and then adding the square of the coefficient ; multiply the equation by 4 times the coefficient of x2, and then add the square of the coefficient, which x had before multiplying.

NOTE. If the equation contain two powers of the unknown quantity, and the exponent of the one is double that of the other, it may be resolved like a quadratic.

SOLUTION OF QUESTIONS.

WHEN a question is proposed, the analyst ought to form a clear idea of its nature, and then attempt to express its terms, and the relations of its parts, in algebraical characters, putting the letters x, y, z, &c. for the unknown quantities in it; and in this way he must deduce as many independent equations from the conditions of the question as there are unknown quantities in it, which he can always do when the question is properly limited; after which, these equations being resolved by the preceding rules, will give the answer or answers.

Suppose the greatest unknown quantity, y the next, z, v, &c. the lesser ones in their order.

Suppose it to be a condition of the question, that

The two quantities together, or their sum, amounts to 18. This condition may be expressed thus,

Their excess, difference, &c. is 6,

Their product, rectangle, the one into the other, or multiplied by it, is 72,

One of them taken out of the other, divided by it, applied to it, or their quotient, is 2,

The greater is to the less, or their ratio is as 4 to 2,

This proportion, by multiplying the means
and the extremes, becomes an equation,
The sum of their squares is 180,
The difference of their squares is 108,

x+y=18 x-y= 6

xy=72

y

= 2

xy:: 4:2

2x=4y

x2+ y2=180 x2-y2 = 108

And in a similar way may any other relations of the quantities be expressed in equations.

When the relation of one unknown quantity to another is simple, a letter may be taken for one of them, and an expression for the other deduced from the relation between them, which will abridge the work, and render it more elegant. Thus, if their difference be 3, take y for the less, and y+3 will be the greater.

It will often abridge the work, if letters are taken not for the unknown quantities themselves, but for their sum, difference, or any other relation from which the quantities may be easily found.

QUESTIONS PRODUCING SIMPLE EQUATIONS.

1. To find such a number, that, if it be multiplied by 5, and also by 3, the former product shall exceed the latter by 26. The first product is 5x, the second 3x, their difference 26.

5x-3x=

Ans. 13.

2. To find a number, to which if 27 be added, the sum shall be 10 times the number required.

10x=x+27.

Ans. 3.

3. To find a number, from which if 4 be taken, and the remainder multiplied by 3, the product shall be twice the number sought. (x-4)x3=2x. Ans. 12. 4. To find a number of which the fourth part exceeds the fifth part by 13.

5. To find a number, to the half of which if 7 be added,

the sum shall be equal to twice the number with 20 taken from it.*

Ans. 18.

6. To find a number of which the square shall be equal to 4 times the number, together with 5 times the same number. Ans. 9.

7. To find a number, to which if its half, its third, and its fourth parts be added, the sum shall be equal to the square of that number.

8. To find a number, from which if 3 be taken, and the remainder multiplied by 3, and then 4 added to the product, the sum divided by 5 shall give half the number sought.

Ans. 10.

9. To find a number of pounds, to which if 3 be added, and the sum multiplied by 12, the product shall be equal to the number of shillings in the value of the pounds, diminished by as many crowns as there are pounds required. Ans. £12. 10. To find two numbers, of which the sum is 133, their difference 47.

and y, and y+47, the numbers, 2y+47-133. Ans. 90 and 43. 11. To find two numbers of which the sum is 84, and their quotient 13. Ans. 78 and 6. 12. To find two numbers of which the difference is 104, and their quotient 9. Ans. 117 and 13. 13. To find two numbers, so that 3 times the greater added to twice the less shall make 54, and 4 times the greater with 3 times the less shall make 75. Ans. 12 and 9. 14. To find two numbers, so that the greater with half the less shall make 25, and the less with half the greater shall make 23. Ans. 18 and 14. 15. To find two numbers in the ratio of 4 to 3, so that if one be added to each of them, the sums shall be in the ratio of 9 to 7.

3x=4y, (x+1)×7=(y+1) × 9.

Ans. 8 and 6.

* The intention of this section is to assist the learner in transferring the conditions of the question from common language into algebraic expressions, and thus forming equations, which are to be solved by the three preceding sections. The equations were inserted in the first edition, as being the proper answers aimed at; but many eminent teachers have suggested that this has a tendency to prevent students from exerting their own powers. They are now therefore omitted, except where some difficulty is apprehended in forming them, and which might not easily be got over without assistance.