Conjugate (square roots) GudangMovies21 Rebahinxxi LK21

    In mathematics, the conjugate of an expression of the form



    a
    +
    b


    d




    {\displaystyle a+b{\sqrt {d}}}

    is



    a
    −
    b


    d


    ,


    {\displaystyle a-b{\sqrt {d}},}

    provided that





    d




    {\displaystyle {\sqrt {d}}}

    does not appear in a and b. One says also that the two expressions are conjugate.
    In particular, the two solutions of a quadratic equation are conjugate, as per the



    ±


    {\displaystyle \pm }

    in the quadratic formula



    x
    =



    −
    b
    ±



    b

    2


    −
    4
    a
    c




    2
    a





    {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}

    .
    Complex conjugation is the special case where the square root is



    i
    =


    −
    1


    ,


    {\displaystyle i={\sqrt {-1}},}

    the imaginary unit.


    Properties


    As




    (
    a
    +
    b


    d


    )
    (
    a
    −
    b


    d


    )
    =

    a

    2


    −

    b

    2


    d


    {\displaystyle (a+b{\sqrt {d}})(a-b{\sqrt {d}})=a^{2}-b^{2}d}


    and




    (
    a
    +
    b


    d


    )
    +
    (
    a
    −
    b


    d


    )
    =
    2
    a
    ,


    {\displaystyle (a+b{\sqrt {d}})+(a-b{\sqrt {d}})=2a,}


    the sum and the product of conjugate expressions do not involve the square root anymore.
    This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is:







    a
    +
    b


    d




    x
    +
    y


    d





    =



    (
    a
    +
    b


    d


    )
    (
    x
    −
    y


    d


    )


    (
    x
    +
    y


    d


    )
    (
    x
    −
    y


    d


    )



    =



    a
    x
    −
    d
    b
    y
    +
    (
    x
    b
    −
    a
    y
    )


    d





    x

    2


    −

    y

    2


    d



    .


    {\displaystyle {\frac {a+b{\sqrt {d}}}{x+y{\sqrt {d}}}}={\frac {(a+b{\sqrt {d}})(x-y{\sqrt {d}})}{(x+y{\sqrt {d}})(x-y{\sqrt {d}})}}={\frac {ax-dby+(xb-ay){\sqrt {d}}}{x^{2}-y^{2}d}}.}


    Hence:






    1

    a
    +
    b


    d





    =



    a
    −
    b


    d





    a

    2


    −
    d

    b

    2





    .


    {\displaystyle {\frac {1}{a+b{\sqrt {d}}}}={\frac {a-b{\sqrt {d}}}{a^{2}-db^{2}}}.}


    A corollary property is that the subtraction:




    (
    a
    +
    b


    d


    )
    −
    (
    a
    −
    b


    d


    )
    =
    2
    b


    d


    ,


    {\displaystyle (a+b{\sqrt {d}})-(a-b{\sqrt {d}})=2b{\sqrt {d}},}


    leaves only a term containing the root.


    See also


    Conjugate element (field theory), the generalization to the roots of a polynomial of any degree

Kata Kunci Pencarian:

conjugate square rootsmultiplying conjugate square rootsconjugate in math with square roots
Complex Conjugate Roots - Examples and Practice Problems - Neurochispas

Complex Conjugate Roots - Examples and Practice Problems - Neurochispas

Conjugate Roots Theorem Calculator Online

Conjugate Roots Theorem Calculator Online

Conjugate

Conjugate

Conjugate

Conjugate

Conjugate of Square Root - Definition and Examples

Conjugate of Square Root - Definition and Examples

Conjugate of Square Root - Definition and Examples

Conjugate of Square Root - Definition and Examples

Conjugate roots - overview | Numerade

Conjugate roots - overview | Numerade

Conjugate roots - overview | Numerade

Conjugate roots - overview | Numerade

Solved 1. When do we use the conjugate method? I thought we | Chegg.com

Solved 1. When do we use the conjugate method? I thought we | Chegg.com

ShowMe - Conjugate Root Theorem

ShowMe - Conjugate Root Theorem

ShowMe - Conjugate Root Theorem

ShowMe - Conjugate Root Theorem

Conjugate In Math - Cuemath

Conjugate In Math - Cuemath