Hence, the numbers are 3 and 4.

2. The sum of two numbers is 17 and their product is 72. Find the numbers.

3. Find two numbers whose difference is 6 and whose product is 27.

4. Find two numbers whose sum is 9 and the difference of whose squares is 9.

5. The sum of the cubes of two numbers is 72 and their sum is 12. Find the numbers.

6. Find the dimensions of a rectangle whose perimeter is 44 ft. and whose area is 120 square feet.

7. The area of a rectangle is 1200 square feet and the length of the diagonal is 50 feet. Find the length and width.

8. Find the dimensions of a rectangular garden of half an acre, which requires 36 rods of fence.

9. A rectangular field having an area of 120 square rods is surrounded by a road 1 rod wide. The area of the road is 48 square rods. Find the dimensions of the field.

10. Two square gardens have together an area of 7400 square feet. A rectangular garden, whose dimensions are equal to the sides of the two squares, requires 240 feet of fence. Find the sides of the two squares.

11. A number is composed of two digits. The sum of the squares of the two digits is 73, and if 18 is added to the

number, the order of the digits will be reversed. Find the number.

12. The sum of the numerator and denominator of a fraction is 7. If the numerator is increased by 1 and the denominator diminished by 1, the resulting fraction will be the reciprocal of the original fraction. Find the fraction.

13. A tailor bought cloth for $400. Had he paid $1 more per yard, he would have received 20 yards less. Find how many yards he bought and the price per yard.

14. Two men working together can mow a field in 6 days. One man working alone can mow the field in 5 days less time than the other. In how many days can each alone do the work?

15. The front wheel of a carriage makes 48 revolutions more than the hind wheel in going one mile. If the circumference of each wheel were increased by 1 ft. the front wheel would make only 40 revolutions more than the hind wheel in going one mile. Find the circumference of each wheel.

16. A sum of money at interest amounted to $2120 at the end of one year. Had the principal been $400 more and the rate 1 percent less, the interest would have been the same. Find the principal and the rate of interest.

Part of the

17. A man has a distance of 20 miles to go. distance he walks, and part he rides in a borrowed motor car which runs 4 times as fast as he can walk, the trip taking him 3 hours and 10 minutes. The return trip, being down hill, he makes in 2 hours and 30 minutes, the motor running 2 miles an hour faster and his rate of walking increased one mile per hour. Find the distance he walks and his rate of walking and riding.

18. Two athletes run a 100-yard race. In the first heat, A is beaten by B by 1 seconds. In the second heat, each increases his speed by one yard per second and A is beaten by 14 seconds. Find the speed at which each ran.

GRAPHICAL SOLUTION OF SIMULTANEOUS QUADRATICS

149. As in the graphical solution of simultaneous simple equations, the values of x and y that will satisfy both equations are the coördinates of the points that lie in both graphs, that is, the coördinates of the points of intersection. If the graphs do not intersect, the equations have no real roots. x2 + y2 25 (1)

=

1. Solve

18 (2)

If x were greater than 5 or less than -5, y would be imaginary. Fig. 1 shows that the graph of (1) is a circle.

The graph of (2) is a straight line, touching the circle at one point only, when x = 3 and y = 4. Hence these two values are the only real roots of these equations.

In 1, if x = Oy is imaginary, since y = ±

when x is less than ± 4, y is imaginary.

x = ± 4,

5x2-80

8,

y = ±

For larger values of x, y will become larger, and the graph of

, any value of x greater than + 8 or less

8, will make y imaginary; hence the graph must lie between

There are 4 points of intersection, hence the roots are x = 4.5

The graph of (1) is a circle found as in Example 1. Fig. 3.

In this case it is more convenient to assume values

For larger values of y, we get larger values of x which bring the graph beyond the circle.

The graph of (2) is a parabola.

In this case, there are only two points of intersection, giving two values of y and one value of x.

When x = 4.5 approx.. Y = 3.8 approx.

The student will note that every quadratic equation will yield one of the curves known as the conic sections, that is a circle, parabola, ellipse or hyperbola.

The position of these curves, with reference to the axes of x and y, may be shifted, depending upon the form of the equations.