I. \(\sqrt{x} + x + 1 = 0\) has two irrational roots.
II. \(5\sqrt{x} - x - 4 = 0\) has two rational roots.
Which of the statements given above is/are correct ?
II only
Let's analyze the given statements to identify which one is correct.
- Statement I: \(\sqrt{x} + x + 1 = 0\) has two irrational roots.
- Consider the equation \(\sqrt{x} + x + 1 = 0\).
- Rearrange it: \(\sqrt{x} = -x - 1\).
- This implies that \(x\) must be non-negative, as it is under the square root.
- For real numbers, both sides must be equal and valid, which is not possible here since the left side is non-negative and the right side is negative for any real number \(x \geq 0\).
- Therefore, there are no real solutions to this equation, let alone irrational roots.
- Statement II: \(5\sqrt{x} - x - 4 = 0\) has two rational roots.
- Consider the equation \(5\sqrt{x} - x - 4 = 0\).
- Rearrange it: \(5\sqrt{x} = x + 4\).
- Square both sides to eliminate the square root: \(25x = x^2 + 8x + 16\).
- Rearranging gives a quadratic equation: \(x^2 - 17x + 16 = 0\).
- To determine the nature of the roots, calculate the discriminant, \(D = b^2 - 4ac\).
- Here, \(a = 1, b = -17, c = 16\), thus \(D = (-17)^2 - 4 \times 1 \times 16 = 289 - 64 = 225\).
- As \(D = 225 \gt 0\) and a perfect square, the quadratic has two distinct rational roots.
Hence, based on the analysis, the correct answer is II only.
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