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264. In the solution of certain equations, it frequently becomes necessary to extract the square root of binomial surds in the form of

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265. To obtain a rule for the operation, when possible,

assume

√a+√b=√x+ √ÿ,

(1)

where one or both terms of the second member are irrational. But by Art. 263,

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Squaring equation (1), we obtain

a+√b=x+2√xy+y;

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Adding and subtracting (3) and (4), and factoring,

(2)

(3)

(4)

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Let, now, a2

b be a perfect square, whose root is c;

and substituting in (5) and (6), we have

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Substituting these values of x and y in (1) and (2), we obtain

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Hence, the square root of a binomial of the form of ab may be found, when a2-b is a perfect square, and (7) and (8) are formulas which apply to any particular example, by substituting the particular values for a, b, and c.

EXAMPLES.

1. Required the square root of 3 + 2√2.

Here a 3, Noī = 2√2 = √8, or b = 8, and

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2. Required the square root of 11+72.

Ans. 3+2.

3. Required the square root of 6+20.

Ans. 1+5.

4. Required the square root of 14- 3. Ans. 23.

5. Required the square root of 43 158.

Ans. 532.

6. Required the square root of 12+2

35. Ans. √5+√7.

7. Required the square root of 2 m +2 / m2 — n2. Ans. m + n + √ m

8. Required the square root of — 2 √√ — 1.

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n.

Here a is equal to 0, and the required root is 1-√√ — 1.

a2 b

9. Required the square root of +1⁄2✔a2 - b2.

4 2

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266. RADICAL EQUATIONS are those containing radical quantities.

267. The solution of a radical equation consists in rationalizing the terms containing the unknown quantity, and in determining its value.

The following examples containing radical quantities reduce to simple equations.

1. Given

+1 = √x−3 +2, to find the value of x.

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Transposing and uniting, √ √ x − 3 + 1

Squaring,

Or,

Transposing,

Dividing by 2,
Squaring,
Whence,

=

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Clearing of fractions, x+42√x+152 = x+34√x+168

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Multiplying both terms of the fraction by ✔ 1 + x + 1,

Or,

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2 + x + 2 √ x + 1 = x + 6

Transposing and uniting, 2x+1 = 4

. Or,

Whence,

x + 1 = 4

x = 3

4. Given {x+(4ax + 17 a2)* } * = x2 + a3, to find

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{ x + (4 a x + 17 a2) 1 } } = x2 + a±

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Simplifying the first member, √x − 1 = 1 + √ ≈ −1

2

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The foregoing illustrations furnish the following general suggestions

1. Transpose the terms of the given equation so that a radical expression may stand alone in one member; then involve each member to a power of the same degree as the radical.

2. If there is still a radical expression remaining, the process of involution must be repeated.

3. Simplify as much as possible before performing the involution.

Radicals may sometimes be removed by multiplying or dividing by a radical expression; hence they sometimes disappear on clearing of fractions. It is also occasionally convenient to rationalize the denominator of a fraction before removing denominators or involving.

EXAMPLES.

6. Given + 6 = √12+x, to find x.

х

Ans. x 4.

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