Part 23

Or, use the sum and product idea developed on the preceding page. The coefficient of x^2 must be unity.

Add the roots and change the sign to get the coefficient of x.

Multiply the roots to get the independent term.

Therefore: The equation is x^2 + 4x - 21 = 0.

In the same way, if the roots are [2 + 3^(1/2)]/7, [2 - 3^(1/2)]/7, the equation is

x^2 - (4/7)x + 1/49 = 0.

Find the sum, the product, and the nature or character of the roots of the following:

1. x^2 - 7x + 12 = 0.

2. 9x^2 - 6x + 1 = 0.

3. x^2 + 2x + 9734 = 0.

4. 16 + 5/x = 17/(x^2).

5. (x - 8)/(x - 3) = x.

6. (x + 7)(x - 6) = 70.

7. x^2 - x(2)^(1/2) = 3.

8. pr^2 + qr + s = 0.

Form the equations whose roots are:

9. 5, -3.

10. 2/3, 5/3.

11. c + d, c - d.

12. -3, -5.

13. [2 +- -3^(1/2)]/5.

14. 8/3 + (2/3)37^(1/2), 8/3 - (2/3)37^(1/2).

15. [-2 +- -2^(1/2)]/2.

16. Solve x^2 - 3x + 4 = 0. Check by substituting the values of x; then check by finding the sum and the product of the roots. Compare the amount of labor required in each case.

17. Solve (x - 3)(x + 2)(x^2 + 3x - 4) = 0.

18. Is e^(4z) + 2e^(3z) + e^(2z) + 2e^z + 2 + e^(-2z) a perfect square?

19. Find the square root (short method): (x^2 - 1)(x^2 - 3x + 2)(x^2 - x - 2).

20. Solve (1.2x - 1.5)/(1.5) + (.4x + 1)/(.2x - .2) = (.4x + 1)/(.5).