Mathematics

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Solve the equation ( 2\sqrt{4x + 1} + x = 38 )

Answer:
To solve ( 2\sqrt{4x + 1} + x = 38 ), follow these steps:

  1. Isolate the square root term:

    Move ( x ) to the other side of the equation:

    2\sqrt{4x + 1} = 38 - x

  2. Square both sides to eliminate the square root:

    (2\sqrt{4x + 1})^2 = (38 - x)^2

    This simplifies to:

    4(4x + 1) = (38 - x)^2

  3. Expand and simplify the equation:

    First, expand the left side:

    16x + 4 = 1444 - 76x + x^2

    Rewrite the equation:

    x^2 - 92x + 1440 = 0

  4. Solve the quadratic equation:

    Use the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ):

    For the quadratic equation ( x^2 - 92x + 1440 = 0 ), ( a = 1 ), ( b = -92 ), and ( c = 1440 ).

    Calculate the discriminant:

    b^2 - 4ac = (-92)^2 - 4(1)(1440) = 8464 - 5760 = 2704

    Find the solutions:

    x = \frac{92 \pm \sqrt{2704}}{2} = \frac{92 \pm 52}{2}

    Therefore:

    x_1 = \frac{92 + 52}{2} = \frac{144}{2} = 72
    x_2 = \frac{92 - 52}{2} = \frac{40}{2} = 20

  5. Verify both solutions:

    • For ( x = 72 ):

      2\sqrt{4(72) + 1} + 72 = 2\sqrt{289} + 72 = 2 \cdot 17 + 72 = 34 + 72 = 106 \neq 38

    • For ( x = 20 ):

      2\sqrt{4(20) + 1} + 20 = 2\sqrt{81} + 20 = 2 \cdot 9 + 20 = 18 + 20 = 38

      ( x = 20 ) is the correct solution.

Final Answer:
The solution to the equation ( 2\sqrt{4x + 1} + x = 38 ) is ( \boxed{x = 20} ).