320. When surds appear in the denominators of fractions in equations, the equations may be cleared of fractional terms by the process described in Art. 186, care being taken to follow the Laws of Combination of Surd Factors given in Art. 305.

321. The following are examples of Surd Equations resulting in quadratics.

Clearing the equation of fractions, 2x+2=5√√x.

Squaring both sides, we get 4x2+8x+4=25x ;

Squaring both sides, x+9=4x-12√x+9;

Hence the values of a which satisfy the equation are 16 and 0 (Art. 248).

A

√(2x2+x)=10−x.

Squaring both sides, 2x2+x=100—20x+x2,

322. We shall now give a set of examples of Surd Equations some of which are reducible to Simple and others to Quadratic Equations.

EXAMPLES.-CXXII.

1. 4x-12√x=16.

{√4. √(6x−11)=√(249 – 2x2).

2. 45-14√x=-x.

5. √(6-x)=2—√(2x-1).

3. 3√(7+2x2)=5√(4x-3). 6. x-2√√(4—3x)+12=0.

~7. √(2x+7)+√(3x-18)=√(7x+1).

8. 2√(204-5x)=20-√√(3x-68).

~11. √(x+5).√(x+12)='12. ≤ 16. √(7x+1)−√(3x+1)=2. 12. √(x+3)+√(x+8)=5√x. 17. √(4+x)+√x=3.

13. √(25+)+√(25-x)=8. 18. √x+√(x+9975)=

525

3

= 5

Jake J

XXVI. ON THE ROOTS OF EQUATIONS.

323. We have already proved that a Simple Equation can have only one root (Art. 193): we have now to prove that a Quadratic Equation can have only two roots.

324. We must first call attention to the following fact:

If mn=0, either m=0, or n=0.

Thus there is an ambiguity: but if we know that m cannot be equal to 0, then we know for certain that n=0, and if we know that n cannot be equal to 0, then we know for certain that m=0.

Further, if lmn=0, then either 1=0, or m=0, or n=0, and so on for any number of factors.

Ex. (1) Solve the equation (x-3)(x+4)=0.

EXAMPLES.-CXXIII.

1. (x-2) (x−5)=0. 2. (x−3)(x+7)=0. 3. (x+9) (x+2) = 0.

4. (x-5α) (x-6b)=0. 6. (19x-227) (14x+83) = 0.

5. (2x+7) (3x-5)=0. 7. (5x-4m) (6x-11n)=0.

8. (x2+5ax+6a2) (x2-7ax+12a3)=0.

9. (x-4)(x2-2ax+a3)=0..

10. x(x-5)=0.

11. (acx-2a+b) (bcx+3a-b)=0.

12. (cx-d) (cx-e)=0. ·

b

a

Writing p for and q for, we may take the following

a

as the type of a quadratic equation of which the coefficient of the first term is unity,

x2+px+q=0.

326. To shew that a quadratic equation has only two roots.

Let x2+px+q=0 be the equation.

Suppose it to have three different roots, a, b, c.