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: 2x2−12x+10=0 - Zero Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: x2+4=4x - Negative Discriminant
Without solving or graphing, determine the nature of the roots of the quadratic equation: x2+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: 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)
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