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of x and y be confined to whole numbers, the values of x will be 1, 2, 3, &c. and the corresponding values of y will be 99, 98, 97, &c.; and that either pair of these numbers will satisfy, or answer the conditions expressed by the equation. And although there is only a certain number of values that x and y can have in this case, yet if we admit x and y to represent fractional as well as integer quantities, this number has no limits, or in other words x and have an infinite number of values. If we now introduce another equation, as for instance x=3y in conjunction with the former one; the values of x and y then become restricted to the numbers 75 and 25 respectively, since no other numbers but these when substituted for x and y will satisfy both equations. If we suppose there to be more equations than unknown quantities, we have more data than required for the determination of the latter, and the equations (if independent ones) will be inconsistent with each other and show some absurdity; therefore such supposition cannot be admitted.

72. By Independent Equations, are meant such equations concerned in a problem that cannot be deduced one from the other by the usual rules of algebra, as addition, multiplication, transposition, &c.; but from the conditions contained in the problem alone. Thus x+y=a and x-y=b

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are Independent Equations, since we cannot by any algebraical process obtain one from the other. On the other hand the equations x+y=a and bx+by=ab, are not independent, inasmuch as the latter is obtained from the former, by multiplying it by the quantity b. The latter equations are therefore identical, that is, they may be considered to be but one and the same equation.

QUADRATIC EQUATIONS.

73. Pure Quadratic Equations are such as, when properly reduced, contain only the square of the unknown quantity, as ax2=p, ax2+b= cx2-d, 25x2-400, &c.

Adfected Quadratic Equations are such as contain the unknown quantity and its square, but in different terms, as ax2+p=dx, x2+2x=10,

&c.

74. RESOLUTION OF PURE QUADRATIC EquaTIONS. -These Equations may be reduced to the general form of x2=a, .. x=±√a, that is, the values of x are +a and -α. (Art. 54.) We must therefore transpose the terms of the Equations, so that the term which contains x2 may be on one side of the Equation, and the known quantities on the other. Reduce the coefficient of x2 to unity, by dividing by that coefficient if necessary (Art. 60); and extract the square root of

both sides of the equation. Such equations contain 2 equal roots or values of x, but with contrary signs, and whether one or both these values will apply to the problem or question under consideration, will depend upon its particular nature, and for which no general rule can be laid down.

Ex. 1. Find x in the equation 2x2-9=23

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2. Find x in the equation ax+p=q

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75. RESOLUTION OF ADFECTED QUADRATIC EQUATIONS. These Equations may be reduced to the simple and general form x2+px=a. If we add

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to each side (Art. 16), the equation

becomes x2+px+=+a. By this contriv

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ance, the left-hand side of the equation is made a Complete Square (since x2+px+ is the square

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ing Rule:-Transpose the terms of the Equation, so that the terms which contain x2 and x, may be on one side of the Equation, and the known quantities on the other, and that the term containing x2 may have a positive sign. Reduce the co

efficient of x to unity (Art. 60); add to each side of the Equation, the square of half the coefficient of x; then extract the square root of both sides of the equation; the result is a simple equation, and 2 values of x become known.

Ex. 1.

Find x in the equation 48-3x2=18x

3x2+18x=48

x2+6x=16

The coefficient of x is 6, and the square of its half is 9, which by the Rule is to be added to both sides of the equation, which then becomes

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completing the square x+4x+4=25

extract square root

x+2=5

x-3 or -7

EXAMPLES IN QUADRATIC EQUATIONS.

76. Find the unknown quantities in the follow

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