Maths-
General
Easy

Question

Solve the radical equation square root of 4 x plus 5 end root minus square root of x plus 1 end root equals 1 Check for extraneous solutions

hintHint:

Rearrange the equation and then solve for x.

The correct answer is: Hence no extraneous solution


    Complete step by step solution:
    Here we have the equation square root of 4 x plus 5 end root minus square root of x plus 1 end root equals 1

    not stretchy rightwards double arrow square root of 4 x plus 5 end root equals 1 plus square root of x plus 1 end root
    On squaring both the sides, we have 4 x plus 5 equals left parenthesis 1 plus square root of x plus 1 end root right parenthesis left parenthesis 1 plus square root of x plus 1 end root right parenthesis
    not stretchy rightwards double arrow 4 x plus 5 equals 1 plus 2 square root of x plus 1 end root plus x plus 1
    On subtracting x and 2 on both the sides, we get
    not stretchy rightwards double arrow 3 x plus 3 equals 2 square root of x plus 1 end root
    On squaring both the sides, we get
    9 x squared plus 18 x plus 9 equals 4 x plus 4
    not stretchy rightwards double arrow 9 x squared plus 14 x plus 5 equals 0
    Now, this is a quadratic equation with a = 9, b = 14, c = 5
    Roots can be found with, x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction
    x equals fraction numerator negative 14 plus-or-minus square root of 196 minus 4 cross times 9 cross times 5 end root over denominator 2 cross times 9 end fraction
    On solving, we get x equals negative 1 comma fraction numerator negative 5 over denominator 9 end fraction
    Here, on substituting x equals negative 1 text  or  end text fraction numerator negative 5 over denominator 9 end fraction in given equation we get
    square root of 4 cross times negative 1 plus 5 end root minus square root of negative 1 plus 1 end root equals 1
    not stretchy rightwards double arrow square root of 1 minus square root of 0 equals 1
    Hence x = - 1 works.
    Now, square root of 4 cross times negative 5 over 9 plus 5 end root minus square root of fraction numerator negative 5 over denominator 9 end fraction plus 1 end root equals 1
    not stretchy rightwards double arrow square root of 25 over 9 end root minus square root of 4 over 9 end root equals 5 over 3 minus 2 over 3 equals 3 over 3 equals 1
    Here x equals negative 5 over 9  also works.
    Hence no extraneous solution

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