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Solution. This equation, reduced as in arts. 85 and 88, is

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√(2x+7) + √(3x-18) = √(7x + 1).

Solution. This equation, being freed from radical signs,

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Examples of Equations of the Second Degree.

Solution. If we proceed to eliminate y between these two equations, by the process of art. 116, the remainder of the first division is

(x2-6x+5) y2 - (10x260x+50)y+24x2-144x+120, in which

x2-6x+5

4

is a factor of each of the coefficients of y, and y2, and of the terms which do not contain y.

Before suppressing this factor, we must see whether, as in art. 118, it may not be equal to zero, in which case we have

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is substituted in the given equations, each of them becomes

у3 - 5 y2 + 6 y = 0,

which is satisfied by the value

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is substituted in the given equations, each of them becomes

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which is the same as the preceding equation, and gives

therefore the same values of y.

Examples of Equations of the Second Degree.

Having thus obtained all the roots of the given equation corresponding to

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we may omit this factor of the above remainder, and it be

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and as this does not contain x, it is unnecessary to proceed farther in the elimination of y, but we may obtain the roots of the equation

which are

y2-10 y + 24 = 0,

y = 4, and y = 6,

and substitute them in the given equation to obtain the corresponding values of x.

Thus, if the value

is substituted in the given equations, each of them becomes

y = 6

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is substituted in the given equations, each of them becomes

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x=5, or = 1, in either of which cases, y = 0, or = 2, or = 3;

or x=(24±√171), in which case, y 6;

or x = 2,

in which case, y=4.

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3(112 - 8x) = 19 + √(3x + 7).

Ans. x = 6, or = 11,8368.

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13. What two numbers are they, whose sum is 32, and

product 240?

Ans. 12 and 20.

14. What two numbers are they, whose sum is a, and product b?

Ans.a+√(a2 - b), and a-√(a2-6).

In what case would the values of these unknown quantities be imaginary?

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that is, the product of two numbers cannot be

greater than the square of half their sum.

15. What two numbers are they, whose difference is 5,

and product 24?

Ans. 5 and 3; or

3 and 5.

16. What two numbers are they, whose difference is a,

and product b? Ans.

2

a±√(b++ a2), and -a±√(b+a2).

17. Find a number, whose square exceeds it by 306.

Ans. 18, or - 17.

18. A person being asked his age, answered, 'My mother was 20 years old when I was born, and her age multiplied

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