Solving this quadratic in the usual way, we find

Substituting these values in the second of the given equations

The solutions of the equations are therefore (x=1, y=2) and (x=—}, y=13). Ans.

CHECK.

4x12-1x2=2, [4(-)-(-3)()=1+11=Y,

SOLUTION. These equations may be first solved for x2 and y2. Thus, adding the equations gives 4 x2=16, and therefore x2=4. Whence, from the second equation we have y2=1.

Since x2-4 and y2=1 it follows that x=±2 and y=±1.

Thus, taking into account all the possible combinations of signs, we get four solutions as follows:

(x=2, y=1); (x=−2, y=1); (x=2, y=−1); (x=−2, y=−1). Ans. CHECK. Substitution of any one of these pairs of values in the given equations shows at once that the equations are then satisfied.

EXERCISES-APPLIED PROBLEMS

1. The sum of two numbers is 3 and the sum of their squares is 5. What are the numbers?

2. A piece of wire 24 inches long is bent into the form of a right triangle whose hypotenuse is 10 inches. What are the lengths of its sides? Work by using two unknown letters, x and y.

3. It takes 52 rods of fence to inclose a rectangular garden containing 1 acre. How long and how wide is the garden?

4. The area of a right triangle is 150 square feet and its hypotenuse is 25 feet long. How long are the sides?

5. The area of a rectangular garden is 1200 square feet and the diagonal across it measures 50

feet. What are the length and breadth?

6. The mean proportional between two numbers is 21 and the A sum of their squares is 58. Find

the numbers.

7. Figure 88 shows two circles just touching (tangent to) each other, the smaller one being out

FIG. 88.

B

side the larger one. If their combined areas (shaded in the figure) are 22 sq. ft. and the line AB which passes through the centers

FIG. 89.

measures 6 feet, what is the radius of each circle? (3+5) ft. and (3-√5) ft. Ans.

8. In Fig. 89, the inner circle is tangent to the outer one internally. If the shaded D area is 47 sq. ft., and AB=6 ft., what is the radius of each circle?

9. Do positive integers exist differing by 3 and such that the sum of their squares is 117? If so, find them.

10. Answer Ex. 9 in case the sum of the squares is taken to be 120, other conditions remaining the

same.

11. A rectangular swimming pool together with a platform around it 25 feet wide covered 37,500 square feet. The area of the platform was that of the pool. What were the dimensions of the pool?

12. Two men working together can complete a piece of work in 3 days. It would take one man 1 day longer than the other to do the work alone. In how many days can each man do the work alone?

[HINT. See Ex. 24, p. 172.]

13. A sum of money on interest for one year at a certain rate brought $7.50 interest.

Κ

A

FIG. 90.

HG

If the rate had been 1% less and the principal $25 more, the interest would have been the same. Find the principal and

rate.

14. The figure shows a large square with two equal-sized smaller squares placed at opposite sides. If the shaded area is 72 square feet and the total perimeter of the figure is 40 feet, find the length of side of each square.

15. Solve Ex. 14 in case there are four small squares placed around the large square, one on each side, other conditions remaining the same as before.

16. If, in Fig. 91, the perimeter is 16 ft., and the area is 7 sq. ft., what is the length of side of each square? Compare Ex. 14.

17. A man traveled 30 miles. If his rate had been 5 miles an hour more, he could have made the journey in 1 hour less time. Find L his time and rate.

A's

18. A and B each traveled 100 miles. speed was 5 miles an hour faster than B's and he arrived at the end of the journey 1 hour ahead of B. What was the rate of each?

FIG. 91.

B

19. The figure shows a semicircle resting upon its base AB. At a certain point P on AB the perpendicular PG measures 4 inches, while at the point Q, which is 1 inch

from P, the perpendicular QF meas

ures 3 inches.

base AB.

G

[HINT. Let x=. AP, y=PB.

and y, then take their sum.

p. 197.]

20. Show that the formulas for the

B

FIG. 92.

length and width of the rectangle whose perimeter is a and whose area is b are

(a+Va2-16b) and (a-Va2-166).

1. What is meant by the square root of a number?

2. How many square roots does a number have? Illustrate your answer by stating the square roots of 64; of 13.

3. State (by means of the tables) the approximate values of √30, √72, √84, √136.

4. How many terms has the square of a binomial?

5. Supply a third term to a2+2 ab that shall make the resulting trinomial a perfect square; to a2+b2; to 2 ab+b2.

6. Complete the square in each of the following expressions:

(a) x2-12x. (c) x2+7x.

(b) x2+8x. (d) x2-3 x.

(e) x2-6x4.

(f) 9 x2+6x.

(g) 25 x6–40 x3.

(h) 16 x4-16 x2.

7. Find the square root of 9 x2-30 ax−3 a2x+25 a2+5 a3+‡ aa.

8. Find the square root of

9 a6-12 a5-26 a1+44 a3+9 a2-40 a+16.

9. Find the square root of 494,209; of 57,686.4324.

10. What is a radical? Give three illustrations.

11. Write the radical expressing the 12th root of 125; the 30th root of 27; the nth root of a.

12. What is meant by the index of a radical? by the radicand? 13. Is √25 a radical? is 25 a radical? Give reason in each

case.

14. What Axiom is used to get rid of the radical in solving an equation containing a radical sign?

15. Solve the equations

√2x+1+7=x.

√2x+5-√x-1=2.

√x+2−√2x-10= √3 x −20.

16. When is a radical in its simplest form?

3

√2' 12' √27 x √x+1' √3-√2

21. What is a quadratic equation? How is it different from a linear equation? Give illustrations of each.

22. Define and illustrate a pure quadratic.

23. Define and illustrate an affected quadratic.

24. What is the principle used in solving quadratic equations by factoring?

25. What kind of quadratic requires that its square be " pleted" before solving?

com

26. Describe accurately the method which you use for " pleting the square" in solving quadratics.

com

27. Solve each of the following equations by factoring and check

your answers.

(a) x2-5x-6=0.

(b) x2-22 x = 23.

(c) x2-2x=4x+55.

(d) x2-x-2=0.