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Nature of roots of quadratic equations: The discriminant

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Now Playing:Nature of roots of quadratic equations the discriminant– Example 1
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  1. Positive Discriminant
    Without solving or graphing, determine the nature of the roots of the quadratic equation: 2x2−12x+10=02x^2-12x+10=0
    Characteristics of quadratic functions
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    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=0a{x^2} + bx + c = 0
    quadratic formula: x=−b±b2−4ac    2ax = \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 b2−4acb^2 - 4ac > 0 : 2 solutions (2 real solutions)
    if b2−4acb^2 - 4ac = 0 : 1 solution (1 real solution)
    if b2−4acb^2 - 4ac < 0 : no solution (2 complex solutions)