Surds

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Solve in the exact form.

(sqrt of 4x+1)+(sqrt of x+1)=2

Someone showed me to do this next:
Square both sides..so..
4x+1+2((sqrt of 4x+1)•(sqrt of x+1))=4
I do not understand where the 2 come from ..and why do we need to multiply the sqrt of 4x+1 and sqrt of x+1 together to get the product.......??????????

  • Surds -

    Whoops I meant ..
    Sqrt of x+3, not x+1..

  • Surds -

    (a+b)^2 = a^2 + ab + b^2

  • Surds -

    a+b)^2 = a^2 + 2ab + b^2
    (sqrt of 4x+1)+(sqrt of x+3)=2
    4x+1 + 2 (sqrt (x+3)sqrt(4x+1)) + x+3=4
    5x+4 +2 (sqrt (5x+4)=4
    5x+2sqrt(5x+4)=0
    5x=-2(sqrt(5x+4)
    square both sides
    25x^2=4(5x+3)
    25x^2-20x-12=0
    (5x+2)(5x-6)=0

    x=-12/5 or x=6/5

    check that.

  • Surds -

    Hmmm. I get

    √(4x+1)+√(x+3)=2
    4x+1 + 2√(4x^2+13x+3) + x+3 = 4
    5x = -2√(4x^2+13x+3)
    25x^2 = 4(4x^2+13x+3)
    25x^2 = 16x^2 + 52x + 12
    9x^2 - 52x - 12 = 0
    (9x+2)(x-6) = 0
    x = -2/9 , 6

    But x=6 does not satisfy the original equation, so -2/9 is the only solution.

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