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Transposing, and adding similar terms,

√5x+x2=5−x.

Squaring both sides,

5x+x2-25-10x+x2.

Transposing, and adding similar terms,
15x=25; whence x=13.

9. Find the value of x in the Equation

4/6x-9

√/6x-2

4/6x+6 √6x+2

Clearing this Equation of fractions, by multiplying each numerator into the denominator of the other fraction,

24x-6x-18-24x-2/6x-12. Transposing, and uniting similar terms,

√6x=6.

Squaring both sides,

Transposing,
Squaring both sides,

=

6x=36; whence x=6.

10. Find the value of x in the Equation

√x+c+√x−c
√x + c −√x-c

Multiplying both terms of the fraction by √x+c+√x−c, (243...2)

=b.

Clearing the Equation of its fraction,

2x+2√x2-c2

2c

X=

This Equation will give

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2x+2√x2-c2=2bc. 2√x2-c2 =

=b.

4x2-4c2-462c2-8bcx+4x2.

Transposing, and uniting similar terms,

8bcx=462c2+4c2.

2bc-2x.

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COMPLETE EQUATIONS

In which the Unknown Quantity is contained in a
Surd Expression.

32. Find the value of x in the Equation

6+3x+4=11.

Transposing, and uniting similar terms, √3x+4=5.

Squaring both sides,

3x+4=25; whence x=7.

33. Find the value of x in the Equation 24-√9x22+=15.

Transposing, and uniting similar terms, 2x2+9=-9.

Squaring both sides,

2x2+9=81; whence x=±6.

34. Find the value of x in the Equation
20-√x3+40=4.

Transposing, and uniting similar terms,
x+40=-16.

Squaring both sides,

23+40 256.

Transposing, and uniting similar terms,
x3=216; whence x=6.

Then, since this Equation has three roots (255), the other two roots will be found by reducing it to a quadratic by division, (253). Dividing both sides of the Equation x3-216=0 by x-6, we have

x2+6x+36=0, or
x2+6x-36:

from which we shall find x=−3+√−27, or −3−√-27.

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Transposing, and adding similar terms,

21/2x2+x=20-2x.

Squaring both sides,

This Equation will give x=4.

38. Find the value of x in the Equation
4√x+16=7√x+16-x-6.

Transposing, and uniting similar terms,

8x2+4x-400-80x+4x2.

Squaring both sides,

from which we shall find x-9.

−3√x+16=
-3x+16=-x-6.

9x+144x2+12x+36;

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Squaring both sides,

32x3+16x2x2+16x3+64x2.

Dividing both sides of this Equation by x2, we have
32x+16=x2+16x+64;

from which we shall find x=12, or 4.

40. Find the value of x in the Equation

(x-2)2=

x+√(x2-9)
x-√(x2-9)

Rationalizing the denominator (243...2), we have
(x+√(x2−9))2

9

(x-2)2= Extracting the square root of each side of this Equation,

x+√x2-9

3

X- 2

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Rationalizing the denominator (243...2), we shall have
(x+√(x2−16))2

x2-6x+9=

16

Extracting the square root of each side of this Equation,
x+√x2-16

x-3=

4

Clearing this Equation of its fraction,

4x-12=x+√x2—16.

Transposing, and adding similar terms,

x2-16-12-3x.

Squaring both sides,

from which we shall find x=5, or 4.

x2-16-144-72x+9x2;

42. Find the value of x in the Equation,

3/x3+3.7 × (x3-37)3 =64. This Equation may be put under the form

(23+37)3×(μ3+37)3=64.

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Multiplying together the two binomials in the first member (x3+37)=64, (241...3).

From this Equation we shall have

x3=27;

which will give x=3.

The other two values of x (255), will be found by reducing the cubic Equation to a quadratic, (253).

Dividing both sides of the Equation x3-27-0 by x- -3,

x2+3x+9=0, or

x2+3x=-9;

43. Find the value of x in the Equation

x42x2+6=230.

from which we shall find x=—

3.

Equations of a Quadratic Form with reference to a Power or Root of the Unknown Quantity.

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