# Nature of roots of quadratic equations: The discriminant

##### Examples

###### Lessons

**Positive Discriminant**

Without solving or graphing, determine the nature of the roots of the quadratic equation: $2x^2-12x+10=0$**Zero Discriminant**

Without solving or graphing, determine the nature of the roots of the quadratic equation: $x^2+4=4x$**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)

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)

###### Basic Concepts

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