The two imaginary roots of a quadratic cannot be equal,
since ^^ — 4 ac is not zero. They have, however, the same
real parts, and the same imaginary parts with opposite signs,
and are, therefore, conjugate imaginaries, § 284.
The expression i^ — 4 ac is called the discriminant of the
expression ax^ -{■bx + c.
Digitized by CjOOQ IC



QUADRATIC EQUATIONS. 2S5

804. The above cases may be summarized as follows :

Case 1. If ^^ — 4 ac > 0, the roots are real and unequal.
Case 2. If b^ — ^ac = 0, the roots are real and equal.
Case 3. If ^* — 4 ac < 0, the roots are imaginary.
305. By finding the value of i^ — 4 ac we can determine
at once the character of the roots of a given equation.
1. x^-5x + 6 = 0.
Here a = 1, 6 = — 6, c = 6.
62_4ac = 26-24 = l.
The roots are real and unequal, and rational.
2. Sx^-\-7x-l = 0,

Here a = 3, 6 = 7, c = — 1.
The roots are real and unequal, and are both surds.
3. 4aj«-12aj + 9 = 0.
Here a = 4, 6 = — 12, c = 9.
62 -4ac = 144 -144 = 0.
The roots are real and equal.
4. 2a;«- 3a; + 4 = 0.
Here a = 2, 6 = - 3, c = 4.
62-4fljc = 9--32 = -28.
The roots are both imaginary.
5. Find the values of m for which

2mx^-\- (5m-\-2)x-h (4:m-hl)=0

has its two roots equal.
Here a = 2m, 6 = 6m + 2, c = 4m4-l.
