THE THEORY OF QUADRATIC EQUATIONS
Reading edition
I. To find the sum and the product of the roots.
The general quadratic equation is
(1)
Or, (2)
To derive the formula, we have by transposing
Completing the square,
Extracting square root,
Transposing,
Hence,
These two values of x we call roots. For convenience represent them by and
Hence,
Adding, (3)
Also,
Multiplying, (4)
Hence we have shown that
and
Or, referring to equation (2) above, we have the following rule:
When the coefficient of is unity, the sum of the roots is the coefficient of x with the sign changed; the product of the roots is the independent term.
Examples:
1.
Sum of the roots
Products of the roots
2.
Sum of the roots
Product of the roots
3.
Sum of the roots
Product of the roots
II. To find the nature or character of the roots.
As before,
The determines the nature or character of the roots; hence it is called the discriminant.
If is positive, the roots are real, unequal, and either rational or irrational.
If is negative, the roots are imaginary and unequal.
If is zero, the roots are real, equal, and rational.
Examples:
1.
The roots are real, unequal, and irrational.
2.
The roots are imaginary and unequal.
3.
The roots are real, equal, and rational.
III. To form the quadratic equation when the roots are given.
Suppose the roots are 3, -7.
Then, Or,
Multiplying to get a quadratic,
Or,
Or, use the sum and product idea developed on the preceding page. The coefficient of must be unity.
Add the roots and change the sign to get the coefficient of x.
Multiply the roots to get the independent term.
The equation is
In the same way, if the roots are the equation is
Find the sum, the product, and the nature or character of the roots of the following:
1.
2.
3.
4.
5.
6.
7.
8.
Form the equations whose roots are:
9. 5, -3.
10.
11.
12. -3, -5.
13.
14.
15.
16. Solve 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
18. Is a perfect square?
19. Find the square root (short method):
20. Solve
21. The glass of a mirror is 18 inches by 12 inches, and it has a frame of uniform width whose area is equal to that of the glass. Find the width of the frame.