The roots of the equation $2\sqrt{x} + 2 \dfrac{1}{\sqrt{x}} = 5$ can be found by solving:
- $16x^2 - 92x + 1 = 0$
- $4x^2 - 25x + 4 = 0$
- $4x^2 - 17x + 4 = 0$
- $2x^2 - 21x + 2 = 0$
- $4x^2 - 25x - 4 = 0$
Reveal answer
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C
Correct answer
Explanation
Let u = sqrt(x). 2u + 2/u = 5 => 2u^2 - 5u + 2 = 0. Roots for u are 2 and 1/2. So sqrt(x) = 2 or 1/2. x = 4 or 1/4. The quadratic equation with these roots is (x - 4)(x - 1/4) = 0 => x^2 - 4.25x + 1 = 0 => 4x^2 - 17x + 4 = 0.
AI explanation
Multiply the equation 2 times the square root of x plus 2 divided by the square root of x equals 5 by the square root of x to clear the denominator, yielding 2x + 2 = 5 times the square root of x. Squaring both sides using the identity (a + b)^2 = a^2 + 2ab + b^2 gives 4x^2 + 8x + 4 = 25x. Subtracting 25x from both sides results in the quadratic equation 4x^2 - 17x + 4 = 0.