9.2 THE DISCRIMINANT

• The number (not including the radical sign) in the quadratic formula is called the , D, of the corresponding quadratic equation, .

 • The discriminant allows you to determine the

nature of the roots of the equation because 

ax2 bx c 0

x b D2a

2 4b ac

discriminant

CASE #1 If , then is a

real number and therefore there are distinct solutions.

 

D 0D 0 D positive

2real

CASE #2 If , then

, so the two solutions from the quadratic formula are both

. We call this a .

 

D 0D 0 D 0 0

2ba

double

root

CASE #3If , then D is negative and would

be an number. So, there are distinct

solutions.  **Note: Imaginary solutions ALWAYS come in

pairs – complex conjugates!

D 0D 0

imaginary2imaginary

If the roots are real, we can also determine if the roots are rational or irrational. If D is a perfect square, the roots are rational. If D is not a perfect square, the roots are irrational.

Find the value of the discriminant and tell how may real and how many imaginary roots the equation has. If it has any real roots, tell whether they are rational.

1. 4x2 5x 30

Find the value of the discriminant and tell how may real and how many imaginary roots the equation has. If it has any real roots, tell whether they are rational.

2. 4x2 4x 3 30

Find the value of the discriminant and tell how may real and how many imaginary roots the equation has. If it has any real roots, tell whether they are rational.

3. 4t 2 4t 5 0

Find the value of the discriminant and tell how may real and how many imaginary roots the equation has. If it has any real roots, tell whether they are rational.

4. 5c2 11c 2 0

Determine the value of k for which the given equation will have exactly one real root.

5. 5c2 kc 8 0

Determine the value of k for which the given equation will have exactly one real root.

6. p2 kp 3 2k 0

Determine the value of k for which the given equation will have two distinct real roots.

7. 5kd 2 4d 2 0