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If 1/(3-"√"5) - (2+"√"5)/(3+"√"5) = a + b"√"c, find the value of a, b and c.

Mathematics

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Answer

a=74a = \frac{7}{4}

b=−34b = -\frac{3}{4}

c=5c = 5

Step-by-step solution

6 steps
  1. Rationalize the first term 13−5\frac{1}{3-\sqrt{5}}.

    To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3−53-\sqrt{5}, which is 3+53+\sqrt{5}.

    13−5=13−5×3+53+5\frac{1}{3-\sqrt{5}} = \frac{1}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} =3+5(3)2−(5)2= \frac{3+\sqrt{5}}{(3)^2 - (\sqrt{5})^2} =3+59−5= \frac{3+\sqrt{5}}{9 - 5} =3+54= \frac{3+\sqrt{5}}{4}
  2. Rationalize the second term 2+253+5\frac{2+2\sqrt{5}}{3+\sqrt{5}}.

    To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3+53+\sqrt{5}, which is 3−53-\sqrt{5}.

    2+253+5=2+253+5×3−53−5\frac{2+2\sqrt{5}}{3+\sqrt{5}} = \frac{2+2\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} =(2+25)(3−5)(3)2−(5)2= \frac{(2+2\sqrt{5})(3-\sqrt{5})}{(3)^2 - (\sqrt{5})^2}

    Expand the numerator:

    (2+25)(3−5)=2(3)−2(5)+25(3)−25(5)(2+2\sqrt{5})(3-\sqrt{5}) = 2(3) - 2(\sqrt{5}) + 2\sqrt{5}(3) - 2\sqrt{5}(\sqrt{5}) =6−25+65−2(5)= 6 - 2\sqrt{5} + 6\sqrt{5} - 2(5) =6+45−10= 6 + 4\sqrt{5} - 10 =−4+45= -4 + 4\sqrt{5}

    The denominator is 9−5=49-5=4.
    So, the second term becomes:

    −4+454=4(−1+5)4\frac{-4+4\sqrt{5}}{4} = \frac{4(-1+\sqrt{5})}{4} =−1+5= -1+\sqrt{5}
  3. Substitute the simplified terms back into the original equation.

    3+54−(−1+5)=a+bc\frac{3+\sqrt{5}}{4} - (-1+\sqrt{5}) = a+b\sqrt{c} 3+54+1−5=a+bc\frac{3+\sqrt{5}}{4} + 1 - \sqrt{5} = a+b\sqrt{c}
  4. Combine the terms on the left-hand side.

    Find a common denominator, which is 4.

    3+54+44−454=a+bc\frac{3+\sqrt{5}}{4} + \frac{4}{4} - \frac{4\sqrt{5}}{4} = a+b\sqrt{c} 3+5+4−454=a+bc\frac{3+\sqrt{5} + 4 - 4\sqrt{5}}{4} = a+b\sqrt{c}

    Combine the rational parts and the irrational parts:

    (3+4)+(5−45)4=a+bc\frac{(3+4) + (\sqrt{5} - 4\sqrt{5})}{4} = a+b\sqrt{c} 7−354=a+bc\frac{7 - 3\sqrt{5}}{4} = a+b\sqrt{c}
  5. Separate the rational and irrational parts to match the form a+bca+b\sqrt{c}.

    74−354=a+bc\frac{7}{4} - \frac{3\sqrt{5}}{4} = a+b\sqrt{c}

    This can be written as:

    74+(−34)5=a+bc\frac{7}{4} + \left(-\frac{3}{4}\right)\sqrt{5} = a+b\sqrt{c}
  6. Compare the terms to find the values of aa, bb, and cc.

    By comparing the rational parts:
    a=74a = \frac{7}{4}
    By comparing the coefficients of 5\sqrt{5}:
    b=−34b = -\frac{3}{4}
    By comparing the radicands:
    c=5c = 5

    The values are:
    a=74a = \boxed{\frac{7}{4}}
    b=−34b = \boxed{-\frac{3}{4}}
    c=5c = \boxed{5}

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