For problems 1 - 4, use the discriminant to determine the nature of the roots.

1. 2x^2+5x-3=0

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

3. x^2+6x+11=0

4. 9x^2-54x+81=0

The Discriminant = b²-4ac

1.

a=2

b = 5

c=-3

The Discriminant is 49

When a, b and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational unequal.

2.

a=-1

b=1

c=1

The Discriminant is 5

When a, b and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax2 + bx + c = 0 are real, irrational and unequal.

3.

a=1

b=6

c=11

The Discriminant is-8

then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and imaginary.

4.

a=9

b=-54

c=81

The Discriminant is 0

the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and equal.