MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND

1. Solve the equation obtaining the values of the roots correct to three significant figures. (Harvard.)

2. Write the roots of (Sheffield Scientific School.)

3. Solve (Yale.)

4. Solve the equation for x, taking and and verify your result. (Harvard.)

5. Solve

6. Solve (Coll. Ent. Board.)

7. Find all values of x and y which satisfy the equations:

(Mass. Inst. of Technology.)

8. If and represent the roots of find and in terms of p, q, and r. (Princeton.)

9. Form the equation whose roots are and

10. Determine, without solving, the character of the roots of (College Entrance Board.)

11. If prove that (College Entrance Board.)

12. Given Prove that (Sheffield.)

13. The 9th term of an arithmetical progression is the 16th term is Find the first term. (Regents.)

Solve graphically:

1.

2.

3. Find four numbers in arithmetical progression, such that the sum of the first two is 1, and the sum of the last two is -19.

4. What number added to 2, 20, 9, 34, will make the results proportional?

5. Find the middle term of

6. Solve (Princeton.)

7. A strip of carpet one half inch thick and feet long is rolled on a roller four inches in diameter. Find how many turns there will be, remembering that each turn increases the diameter by one inch, and that the circumference of a circle equals (approximately) times the diameter. (Harvard.)

8. The sum of the first three terms of a geometrical progression is 21, and the sum of their squares is 189. What is the first term? (Yale.)

9. Find the geometrical progression whose sum to infinity is 4, and whose second term is

10. Solve

11. Solve

12. Two hundred stones are placed on the ground 3 feet apart, the first being 3 feet from a basket. If the basket and all the stones are in a straight line, how far does a person travel who starts from the basket and brings the stones to it one by one?

Solve graphically; and check by solving algebraically:

1.

2.

3.

Determine the value of m for which the roots of the equation will be equal: (Hint: See page 40. To have the roots equal, must equal 0.)

4.

5.

6. If is a root of find the other root without solving the equation. (Univ. of Penn.)

7. How many times does a common clock strike in 12 hours?

8. Find the sum to infinity of ···.

9. Solve

10. Find the value of the recurring decimal 2.214214....

11. A man purchases a $500 piano by paying monthly installments of $10 and interest on the debt. If the yearly rate is 6%, what is the total amount of interest?

12. The arithmetical mean between two numbers is and their geometrical mean is 42. Find the numbers. (College Entrance Exam. Board.)

13. If the middle term of is equal to the fourth term of find the value of x. (M. I. T.)