Nature of roots of quadratic equations: The discriminant

Nature of roots of quadratic equations: The discriminant

The discriminant is actually part of the quadratic formula. It is super useful when we only need to determine whether a quadratic equation has 2 real solutions, 1 real solution, or 2 complex solutions.

Lessons

• For the quadratic equation: ax2+bx+c=0a{x^2} + bx + c = 0
quadratic formula: x=b±b24ac2ax = \frac{{ - b \pm \sqrt {{b^2} - 4ac\;} \;}}{{2a}}

• discriminant: b² - 4ac
The discriminant (\vartriangle), b² - 4ac, can be used to discriminate between the different types of solutions:
if b24acb^2 - 4ac > 0 : 2 solutions (2 real solutions)
if b24acb^2 - 4ac = 0 : 1 solution (1 real solution)
if b24acb^2 - 4ac < 0 : no solution (2 complex solutions)
  • 1.
    Positive Discriminant
    Without solving or graphing, determine the nature of the roots of the quadratic equation: 2x212x+10=02x^2-12x+10=0

  • 2.
    Zero Discriminant
    Without solving or graphing, determine the nature of the roots of the quadratic equation: x2+4=4xx^2+4=4x

  • 3.
    Negative Discriminant
    Without solving or graphing, determine the nature of the roots of the quadratic equation: x2+x+1=0x^2+x+1=0