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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
5.
Quadratic Equations and Inequalities
5.1
Solving quadratic equations by factorising
5.2
Solving quadratic equations by completing the square
5.3
Using quadratic formula to solve quadratic equations
5.4
Nature of roots of quadratic equations: The discriminant
5.5
Applications of quadratic equations
5.6
Solving quadratic inequalities