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Note 2. In solving equations of the above form, add such a quantity to both members that the expression without the radical in the first member may be the same as that within, or some multiple of it.

4. Solve the equation 2x2+5x-2x√x2+5x-3= 12. The equation may be written

+5 −2xv+b2−3+ =12.

Subtracting 3 from both members,

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10. (3x2+x-1)3 - 26(3x2+x-1)= 27.

11. x2+7+√x2 +7 = 20.

12. (2x2+3x-1)+2x2+3x-3=0.

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25. x+14x3 +71x2 + 154 x + 120 = 0.

26. (x-a)2+2√x(x − a) = a2 −2a√x.

27.

+

x2+4x+1, 22+3x+15
x2+3x+1 x2+4x+1 2

28. 4(x-1)-5(x − 1) ̃3 +1 = 0.
29. 9x-4x2-5+√4x2-9x+11=0.

30. 24x3+94x2+600x2975=0.

=

31. (2x+5)*+31 (2x+5) ̄ = 32. 32. 3x(3-x)=11-4√x2-3x+5. 33. (x− a)3 +2√(x − a)3 — 3b = 0.

34. 9x24x3-65x-108x+140 = 0.

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XXIII. SIMULTANEOUS EQUATIONS.

INVOLVING QUADRATICS.

366. Two equations of the second degree (Art. 179) with two unknown quantities will generally produce, by elimination, an equation of the fourth degree with one unknown quantity; the rules already given are, therefore, not suffi cient for the solution of all cases of simultaneous equations of the second degree with two unknown quantities. Consider, for example, the equations

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or,

x+a2-2ax2+ x1 = b,

which is an equation of the fourth degree.

In several cases, however, the solution of simultaneous equations with two unknown quantities may be effected by means of the rules for quadratics.

Note. On the use of the double signs ± and F.

If two or more double signs are used in a single equation, it will be understood that the equation can be read in two ways; first, reading all the upper signs together; second, reading all the lower signs together.

Thus, the equation ab±c can be read either

a+b= c, or a- b = =-C.

And the equation a±b=‡c can be read either

a+b=c, or abc.

The same notation will be used in the case of two or more equa tions, each involving double signs.

Thus, the equations x=±2, y=±3, can be read either

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And the equations x=±2, y=3, can be read either

x = +2, y = −3, or x=-2, y = + 3.

367. CASE I. When each equation is in the form

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Substituting from (3) in (1), 12 + 4 y2 = 76.

Whence,

y2 = 16.

y = ± 4.

Therefore, x=2, y = ±4; or, x=-2, y = ± 4.

Note. In this case there are four possible sets of values of x and y which satisfy the given equations:

1. x2, y = 4.

2. x=2, y = −4.

3. x2, y = 4.

4. x=2, y = −4.

It would be incorrect to leave the result in the form x=±2, y=4; for, by Art. 366, Note, this represents only the first and fourth of the above sets of values.

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