Affected Quadratic Equations.

Exercise 73.

1. Find two numbers whose sum is $11$, and whose product is $30$.

2. Find two numbers whose difference is $10$, and the sum of whose squares is $250$.

3. A man is five times as old as his son, and the square of the son's age diminished by the father's age is $24$. Find their ages.

4. A number increased by its square is equal to nine times the next higher number. Find the number.

5. The square of the sum of any two consecutive numbers lacks $1$ of being twice the sum of the squares of the numbers. Show that this statement is true.

6. The length of a rectangular court exceeds its breadth by $2$ rods. If the length and breadth were each increased by $3$ rods, the area of the court would be $80$ square rods. Find the dimensions of the court.

7. The area of a certain square will be doubled, if its dimensions are increased by $6$ feet and $4$ feet respectively. Find its dimensions.

8. The perimeter of a rectangular floor is $76$ feet and the area of the floor is $360$ square feet. Find the dimensions of the floor.

9. The length of a rectangular court exceeds its breadth by $2$ rods, and its area is $120$ square rods. Find the dimensions of the court.

10. The combined ages of a father and son amount to $64$ years. Twice the father's age exceeds the square of the son's age by $8$ years. Find their respective ages.

Exercise 74.

Ex. A boat sails $30$ miles at a uniform rate. If the rate had been $1$ mile an hour more, the time of the sailing would have been $1$ hour less. Find the rate of the sailing.

lint Let x = the rate in miles per hour. lint Then 30x = the number of hours.

On the other supposition the rate would have been $x+1$ miles an hour and the time ${\displaystyle \frac{30}{x+1}}$.

lint Hence 30x - 30x + 1 = the difference in hours for the sailing. lint But 1 = the difference in hours for the sailing. 30x - 30x + 1 = 1.

The solution of this equation gives $x=5$, or $x=-6$.

Therefore, the rate of the sailing is $5$ miles an hour.

1. A boat sails $30$ miles at a uniform rate. If the rate had been $1$ mile an hour less, the time of the sailing would have been $1$ hour more. Find the rate of the sailing.