# Nature of roots of quadratic equations: The discriminant

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##### Examples
###### Lessons
1. Positive Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: $2x^2-12x+10=0$
1. Zero Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: $x^2+4=4x$
1. Negative Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: $x^2+x+1=0$
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###### Topic Notes
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.
• For the quadratic equation: $a{x^2} + bx + c = 0$
quadratic formula: $x = \frac{{ - b \pm \sqrt {{b^2} - 4ac\;} \;}}{{2a}}$

• discriminant: b² - 4ac
The discriminant ($\vartriangle$), b² - 4ac, can be used to discriminate between the different types of solutions:
if $b^2 - 4ac$ > 0 : 2 solutions (2 real solutions)
if $b^2 - 4ac$ = 0 : 1 solution (1 real solution)
if $b^2 - 4ac$ < 0 : no solution (2 complex solutions)