Nature of roots of quadratic equations: The discriminant

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Lesson: 2
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Lesson: 3
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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.

Basic Concepts: Quadratic function in general form:

y = ax^2 + bx+c

, Solving quadratic equations using the quadratic formula, Shortcut: Vertex formula, Multiplying and dividing radicals

Lessons

• 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:

b^2 - 4ac

> 0 : 2 solutions (2 real solutions)

b^2 - 4ac

= 0 : 1 solution (1 real solution)

b^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: $2x^2-12x+10=0$

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

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

Nature of roots of quadratic equations: The discriminant

Nature of roots of quadratic equations: The discriminant

Lessons

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