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indicate the negative square root, it is done by the symbol -V .

Thus, V16 means +4, while -V16 means –4. Similarly, Va=a and -Va?=-a.

138. Equations Containing Radical Signs. Equations containing radical signs are often solved by squaring each member. This is equivalent to multiplying each member by the same quantity, and hence is justified by Axiom I (§ 9).

EXAMPLE 1. Solve the equation Vx-2=6.
SOLUTION. Squaring both members gives

x-2=36. Whence,

x=38. Ans. CHECK.

V38–2=V36=6.
EXAMPLE 2. Solve the equation Væ-1-Væ–4=1.
SOLUTION. First, transpose the -V2-4 to the right, giving

Vx-1=1+V2–4. Square both members, using Formula V, p. 101, for finding (1+Vx-4)2. This gives

x-1=1+2·1.V2-4+(Vx-4)2, or,

x-1=1+2Vx-4+x-4. Canceling x from both sides and transposing the 1 and -4 to the left side, gives

2=2V«—4, or Vä-4=1. Whence (squaring again),

x-4=12=1. Therefore

x=5. Ans. CHECK. V5-1-V5–4=V7-V1=2-1=1.

Note. It is especially important to check all answers as above for equations containing radicals, since the process of squaring both members sometimes leads to a new equation whose roots do not all belong to the first one. Thus, if we square both members of the equation x=5 we get x2=25, and this last equation has x=-5 as a root as well as x=5.

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EXERCISES Solve each of the following equations, and check your answer in each case. 1. Væ-10=5.

10. V2 x–3 a2+ V2 x=3 a. 2. 4= Vx-3.

14 Væ+8 Væ+3 3. V2 x+1=3.

V2 x+7 V2 x+2 4. Vx2+5=3.

[HINT. Square both sides, 5. Vx+5– Væ=1.

remembering that the square of

a fraction equals the square of its 6. Væ–2+Vx+3=5.

numerator divided by the square 7. V2 x+5 – V2 x+2=1. of its denominator.] 8. V3 x+7-V2 x+10=0. 1, Va–3 x _ 2V7 9. Vx+a= Væ+a.

| 2Vz Va+3 0 [Hint. This being a literal equation (see § 106) we are to find x in terms of a.]

13. If 10 be added to 2 times a certain number, the square root of the result is 4. What is the number?

[Hint. Form an equation and solve it.]

14. If 9 be added to the square of a certain number, the square root of the result is 5. What is the number?

15. The difference between the square root of a certain number and the square root of 11 less than that number is 1. What is the number?

For further exercises on this chapter, see Appendix, pp. 308–309.

CHAPTER XV

RADICALS

139. Radicals. Suppose we have a square (see Fig. 74) which we know contains exactly 2 sq. ft. How long is each of its four sides? In order to answer this, we let z represent the desired length. Then, we must have

X • x=2, or,

X2=2. Hence,

x =V2 ft. Ans. This number, V2, cannot be found exactly, as explained in $ 135, because it is the square root of a number, 2, which is not a perfect square. Still, v2 measures a perfectly definite length, as indicated in Fig. 74. The value of it, correct to two decimal places only, is, by the process in § 135, found to be 1.41.

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Such a number as V2 is called a radical number, or briefly, a radical. It is an indicated root whose value cannot, in general, be exactly determined.

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LEIBNITZ (Gottfried Wilhelm von Leibnitz, 1646–1716) Celebrated for his interest and ability in all branches of learning, especially in the fields of mathematics and philosophy. A contemporary of Newton and regarded as sharing with him the honor of inventing the Calculus.

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