Solve the equation ( 2\sqrt{4x + 1} + x = 38 )
Answer:
To solve ( 2\sqrt{4x + 1} + x = 38 ), follow these steps:
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Isolate the square root term:
Move ( x ) to the other side of the equation:
2\sqrt{4x + 1} = 38 - x -
Square both sides to eliminate the square root:
(2\sqrt{4x + 1})^2 = (38 - x)^2This simplifies to:
4(4x + 1) = (38 - x)^2 -
Expand and simplify the equation:
First, expand the left side:
16x + 4 = 1444 - 76x + x^2Rewrite the equation:
x^2 - 92x + 1440 = 0 -
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 = 2704Find the solutions:
x = \frac{92 \pm \sqrt{2704}}{2} = \frac{92 \pm 52}{2}Therefore:
x_1 = \frac{92 + 52}{2} = \frac{144}{2} = 72x_2 = \frac{92 - 52}{2} = \frac{40}{2} = 20 -
Verify both solutions:
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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.
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Final Answer:
The solution to the equation ( 2\sqrt{4x + 1} + x = 38 ) is ( \boxed{x = 20} ).