plus the double product of this first term a; by a second, which must

p p . "p p2

be —, since px=2—x; therefore, if the square of — or —, be

added to x3-\-px, the first member of the equation will become the p

square of x+~^; but in order that the equality may not be destroy. f

ed — must be added to the second member. 4

By this transformation, the equation x3-\-px=q becomes , p2 p2

Whence by extracting the square root

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From this we derive, for the resolution of complete equations of the second degree, the following general


After reducing the equation to the form x2-fpx=q, add the square of half of the co-efficient of x, or of the second term, to both members; then extract the square root of both members, giving the double sign =fc to the second member; then find the value of x from the resulting equation.

This formula for the value of x may be thus enunciated.

The value of the unknown quantity is equal to half the co-efficient of x, taken with a contrary sign, plus or minus the square root of the known term increased by the square of half the co-efficient of x. Take, for an example, the equation

5 1 3 2 273


Clearing the fractions, we have

lOz2—6z+9=96—8*—12^+278, or, transposing and reducing,


and dividing both members by 22,

2 360

t1 \2

Add y—J to both members, and the equation becomes 2 /1 \2 360 / 1 \2

whence, .by extracting the square root,

1 /360 7TT2

*+22=±V -W+(22V'


22 v 22 T\22/' which agrees with the enunciation given above for the double value of x.

It remains to perform the numerical operations. In the first 360 / 1 x2

place, +(^J must be reduced to a single number, having

(22)2 for its denominator.

360 / 1 \2_360x22+l_7921 NoW' ~W+\22) ~ {22f =:(22)2;

extracting the square root of 7921, we find it to be 89; therefore,

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1 89

Consequently, ±—.

Separating the two values, we have
1 89_88
X=~ 22+22~2a~4'
1 89 45

Therefore, one of the two values which will satisfy the proposed equation, is a positive whole number, and the other a negative fraction.

For another example, take the equation which reduces to

37 57 37 /37\3

If we add the square of —, or 1 — 1 to both members, it becomes

whence, by extracting the square root

37 . ?~57 73V


12 V 6^V12/
/37\3 57

In order to reduce y—J —— to a single number, wo will observe, that


therefore, it is only necessary to multiply 57 by 24, then 37 by itself, and divide the difference of the two products by (12)". Now, 37x37=1369; 57x24=1368;

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"12 12 12

This example is remarkable, as both of the values are positive, and answer directly to the enunciation of the question, of which the proposed equation is the algebraic translation.

Let us now take the literal equation

4a2-2x2+2ax=18a6-1862. By transposing, changing the signs, and dividing by 2, it becomes x?—ax=2a29ai+9b2; whence, completing the square,

d? 9a2

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1. Find a number such, that twice its square, increased by three times this number, shall give 65.

Let a; be the unknown number, the equation of the problem will be 2x2+3x=65,


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