Part 22

Transposing, x = -b/2a +- [[b^2 - 4ac]^(1/2)]/(2a).

Hence, x = [-b +- [b^2 - 4ac]^(1/2)]/(2a).

These two values of x we call roots.

For convenience represent them by r1 and r2.

Hence, r1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a). r2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). --------------------------------------------- Adding, r1 + r2 = -(2b)/(2a) = -b/a. (3)

Also, r1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a). r2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a). ------------------------------------------- Multiplying, r1 r2 = (b^2)/(4a^2) - (b^2 - 4ac)/(4a^2) = (b^2 - b^2 + 4ac)/(4a^2) = (4ac)/(4a^2) = c/a. (4)

Hence we have shown that

r1 + r2 = -b/a, and r1 r2 = c/a.

Or, referring to equation (2) above, we have the following rule:

When the coefficient of x^2 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. x^2 - 9x + 21 = 0. Sum of the roots = 9. Products of the roots = 21.

2. 3x^2 - 7x - 18 = 0. Sum of the roots = 7/3. Product of the roots = -6.

3. -21x = 17 - 4x^2. Sum of the roots = 21/4. Product of the roots = -17/4.

~II. To find the nature or character of the roots.~

As before, r1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a), r2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).

The [b^2 - 4ac]^(1/2) determines the nature or character of the roots; hence it is called the discriminant.

~If b^2 - 4ac is positive, the roots are real, unequal, and either rational or irrational.~

~If b^2 - 4ac is negative, the roots are imaginary and unequal.~

~If b^2 - 4ac is zero, the roots are real, equal, and rational.~

EXAMPLES:

1. x^2 - 4x + 2 = 0.

2. x^2 - 4x + 6 = 0.

3. x^2 - 4x + 4 = 0.

~III. To form the quadratic equation when the roots are given.~

Suppose the roots are 3, -7.

Then, x = 3, Or, x - 3 = 0, x = -7. x + 7 = 0. ------------------- Multiplying to get a quadratic, (x - 3)(x + 7) = 0.

Or, x^2 + 4x - 21 = 0.