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SECTION XII.

EQUATIONS OF THE SECOND DEGREE.

(160.) An equation of the second degree is one in which the highest power of the unknown quantity is a square.

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are equations of the second degree.

(161.) Equations of the second degree are divided into two classes.

I. Equations which contain only the square of the unknown quantity and known terms. These are called incomplete equations.

Of this description are the equations

3x2+12=150; ax2=b, etc.

II. Equations which contain both the first and second powers of the unknown quantity, together with known terms. These are called complete equations. Of this description are the equations

x2-10x=7; ax2+bx=c, etc.

INCOMPLETE EQUATIONS OF THE SECOND DEGREE.

(162.) Every incomplete equation of the second de. gree can be reduced to an equation containing but

two terms.

QUEST.-What is an equation of the second degree? What are in complete equations of the second degree? What are complete equa tions? Every incomplete equation can be reduced to what form?

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Clearing the equation of fractions, we obtain

8x2-72+10x2-7-24x2+299.

Transposing terms, we obtain

24x2+8x2+10x2=7+299+72;

and uniting similar terms, we have

42x2=378;

or, dividing each member by 42,

x=9.

Hence every incomplete equation of the second degree can be reduced to an equation of the form

x2= a;

and, for this reason, incomplete equations are sometimes called equations of two terms.

(163.) If we extract the square root of each member of this equation, we obtain

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Hence, to solve an incomplete equation of the second degree, we have the following

RULE.

Find the value of x2, and extract the square root of both members of the equation.

QUEST.-Why are incomplete equations called equations of two terms? Give the rule for solving an incomplete equation.

Ex. 1. What is the value of x in the equation

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This value may be verified by substitution in the original equation. We thus obtain

or

that is,

5x 4'-18=3x 4'+14,

5x16-18=3x16 +14;
62=62.

It should, however, be observed, that the square root of 16 is either +4 or -4, for -4x-4=16; see Art. 62. And this value may also be verified by substitution in the original equation.

(164.) A root of an equation is the value of the unknown quantity in the equation. The preceding equation has two roots, viz., +4 and -4, and universally we shall find,

1st. Every incomplete equation of the second degree has two roots.

2d. These roots are numerically equal, but have contrary signs.

Ex. 2. What are the values of x in the equation x2-17=130-2x2?

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QUEST. What is a root of an equation? How many roots has an in complete equation of the second degree? What relation have these roots to each other?

Therefore

and

x2=49,
x+7 or -7.

Ex. 3. What are the values of x in the equation

7x2-24-4x2+51?

Ans. x= +5 or -5.

Ex. 4. What are the values of x in the equation

6x2-48-2x2=96?

Ans. x+6 or -6.

Ex. 5. What are the values of x in the equation

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Ex. 6. What are the values of x in the equation

2x2

12+6x=

+52 +15?

3

Ans. x=+3 or -3

Ex. 7. What are the values of x in the equation

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Ex. 8. What are the values of x in the equation

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Ex. 9. What number is that which, being multi

plied by itself, gives the product 256?

Ans. +16 or -16.

Ex. 10. What number is that the third part of whose square being subtracted from 18, leaves a remainder equal to 6?

Ans. +6 or -6.

Ex. 11. A boy, being asked his age, answered that

if it were multiplied by itself, and from the product 108 were subtracted, the remainder would be the square of half his age. What was his age?

Ans. 12 years.

Ex. 12. What two numbers are those whose sum is to the greater as 10 to 7, and whose sum, multiplied by the less, produces 270?

Let 10x represent the sum.

Then 7x will represent the greater number, and 3x will represent the less.

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and the numbers are 21 and 9.

Ex. 13. What two numbers are those which are to each other as 4 to 5, and the difference of whose squares is 81?

Let 4x and 5x represent the numbers.

Ans. 12 and 15.

Ex. 14. What two numbers are those whose difference is to the greater as 2 to 9, and the difference of whose squares is 128?

Let 9x and 7x represent the two numbers.

Ans. 18 and 14.

Ex. 15. What two numbers are those which are to each other as 2 to 3, and the sum of whose squares is 117?

Ans. 6 and 9.

Ex. 16. What two numbers are those whose difference is to the greater as 3 to 8, and the sum of whose squares is 356 ?

Ans. 10 and 16.

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