11.* One Root Known. When one root of a given quadratic equation is known, by applying principle (i.), § 5, the other root may be found immediately without solving the equation.

Ex. 1. Knowing that one root of x2+5x

240 is 3, find the other. Since the coefficient, 5, of x with its sign changed is the sum of the roots, we may obtain the required root, -5, that is, 5-3=-8.

8, by subtracting the given root 3 from

The root may also be obtained by dividing the known term 24 by 3. (Compare with Ex. 2 § 8.)

Ex. 2. Knowing that one root of the equation x2 - 4 x − 1 = 0 is 2- √5, we can find the remaining root by subtracting 2-√5 from the coefficient of x with its sign changed.

We have,+4(2 − √5) = 2 + √5. (Compare with Ex. 5, § 8.) Since in an equation in which the coefficients are rational, irrational or complex roots enter in conjugate pairs, if indeed they enter at all, it follows that we may obtain the required root immediately by writing the binomial 2+ √5, which is the conjugate of the one given, 2 — √/5.

EXERCISE XXIII.

(This exercise may be omitted when the chapter is read for the first time.)

Find, without solving, the remaining root of each of the following equations, when one of the roots is given:

1. x2 → 3 x – 130 = 0, one root being 13.

2. x2 + 2 x −

3. x2 - 21 x

5.

1300, one root being 4. 4x2 + 28 x — 15 = 0, one root being . 5. 182 117 x + 37 = €0, one root being 3. 6. 25 x2 - 85 x 18 = 0), one root being

7. 3x2

8. 98 x2 9. x2

— b.

(a + b)x + 2 ab — 2 b2 = 0, one root being a —

* This section may be omitted when the chapter is read for the first time.

=

10. 2 ax2+(4-ab)x 2 b, one root being b/2.
11. cda (cd)x + 1 = 0, one root being 1/c.
12. 6a2x2+5 ax + 10, one root being 1/2 a.
13. 4x24 ax = b2a2, one root being (a + b)/2.
14. ab cx2-b(ca)x1 = 0, one root being 1/ub.
15. 4 abx2 + 2 (a2 + b2)x + ab
16. acx2+b(a2 + c2)x + ab3c =

=

0, one root being — a/2 b. 0, one root being - ab/c.

18. (28) *.

Express each of the following numbers as a power of 2:

17. (43)2.

Solve each of the following equations:

What roots may possibly be introduced by squaring both members of each of the following equations?

73. (√5 +√− 5)2. 74. (√—7—√7)2. 75. (2 a1+4)2.

Solve each of the following equations:

76. 0=2√√√√x – 7.

77. √xa = 0.

.

78. (3)(9732 165) = 0.

82. (a —b)(c — b) (d — c) (d − a) = (a − b) (b − c) (c — d) (d — a).

84. Realize

85. Find the values of (a + b) and (a2 + b2)°.

Simplify each of the following:

1

2

CHAPTER XXIV

IRRATIONAL EQUATIONS AND SPECIAL EQUATIONS
CONTAINING A SINGLE UNKNOWN

IRRATIONAL EQUATIONS

1. An irrational equation is an equation in which one or more terms are irrational with reference to the unknown.

√x + 5 + √x + 12 = 7.

√x + 4 − 2 √x + 13 + √/x + 22 = 0.

2. It is understood that, when the terms of an equation are af fected by radical signs, principal values only of the roots are to be taken, unless we have means for knowing that other values should be taken. (See Chap. XVIII. § 13, also Chap. XXIV. Ex. 3, § 6, and Ex. 5 and Ex. 6, § 10.)

3. To be consistent it is necessary that a specified letter represent the same numerical value wherever it appears in the terms of a given conditional equation.

E. . g. In the equation x2 - 5x+6= 0, x may represent either 3 in every term or 2 in every term, but never 3 in one term and 2 in another

term.

Thus the equation x2 -5x+6=0 represents either of the following numerical identities:

It is also necessary that a specified symbol of operation, such as √, represent the same root wherever it may appear among the different terms of an equation.

It may be seen that if is to be considered as representing the positive value of the root in one term, to be consistent we must