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Nature of roots of quadratic equations: The discriminant
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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=ax2+bx+c, Solving quadratic equations using the quadratic formula, Shortcut: Vertex formula, Multiplying and dividing radicals
Related Concepts: System of linear-quadratic equations, Graphing quadratic inequalities in two variables, Complex numbers and complex planes
Lessons
• For the quadratic equation: ax2+bx+c=0
quadratic formula: x=2a−b±b2−4ac
• discriminant: b² - 4ac
The discriminant (△), b² - 4ac, can be used to discriminate between the different types of solutions:
if b2−4ac > 0 : 2 solutions (2 real solutions)
if b2−4ac = 0 : 1 solution (1 real solution)
if b2−4ac < 0 : no solution (2 complex solutions)
quadratic formula: x=2a−b±b2−4ac
• discriminant: b² - 4ac
The discriminant (△), b² - 4ac, can be used to discriminate between the different types of solutions:
if b2−4ac > 0 : 2 solutions (2 real solutions)
if b2−4ac = 0 : 1 solution (1 real solution)
if b2−4ac < 0 : no solution (2 complex solutions)
- 1.Positive Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: 2x2−12x+10=0 - 2.Zero Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: x2+4=4x - 3.Negative Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: x2+x+1=0
Do better in math today
18.
Quadratic functions
18.1
Characteristics of quadratic functions
18.2
Graphing parabolas for given quadratic functions
18.3
Finding the quadratic functions for given parabolas
18.4
Solving quadratic equations by factoring
18.5
Solving quadratic equations by completing the square
18.6
Using quadratic formula to solve quadratic equations
18.7
Nature of roots of quadratic equations: the discriminant