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الموضوع: The Quadratic Formula

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    عضو ذهبي الصورة الرمزية زهراء غانم
    تاريخ التسجيل
    Jul 2011
    الدولة
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    936

    افتراضي The Quadratic Formula

    This article is about quadratic equations and solutions. For more general information about quadratic functions, see Quadratic function. For more information about quadratic polynomials, see Quadratic polynomial.

    In mathematics, a quadratic equation is a polynomialequation of the second degree. The general form is
    where x represents a variable, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation becomes a linear equation.)
    The constants a, b, and c, are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below).
    One common use of quadratic equations is computing trajectories in projectile motion. Another common use is in electronic amplifier design for control of step response and stability.


    Plots of real-valued quadratic function ax2 + bx + c, varying each coefficient separately



    التعديل الأخير تم بواسطة زهراء غانم ; 24-07-2011 الساعة 12:39 AM

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  2. #2
    عضو ذهبي الصورة الرمزية زهراء غانم
    تاريخ التسجيل
    Jul 2011
    الدولة
    الاردن - ماركا
    المشاركات
    936

    افتراضي رد: The Quadratic Formula

    A quadratic equation with real or complexcoefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

    The roots are given by the quadratic formula
    where the symbol "±" indicates that both
    are solutions of the quadratic equation.
    Discriminant



    Example discriminant signs
    ■ <0: x2+12
    ■ =0: −43x2+43x13
    ■ >0: 32x2+12x43



    In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case Greek delta, the initial of the Greek word Διακρίνουσα, Diakr&#237;nousa, discriminant:
    A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

    • If the discriminant is positive, then there are two distinct roots, both of which are real numbers:

    For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals.

    • If the discriminant is zero, then there is exactly one distinct real root, sometimes called a double root:


    • If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:

    where i is the imaginary unit.Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
    Monic form


    Dividing the quadratic equation by coefficient a gives the simplified monic form of
    x2 + px + q = 0,where p = ba and q = ca. This in turn simplifies the root and discriminant equations somewhat to
    and
    Δ = p2 − 4q.
    التعديل الأخير تم بواسطة زهراء غانم ; 24-07-2011 الساعة 12:42 AM

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