1.2. Between which two consecutive integers does \( \sqrt{138} \) lie? 1.3. Given \( q=\sqrt{b^{2}-4 a c} \). If \( a=-1 ; b=-2 \) and \( c=4 \). Determine between which two Consecutive integers does \( q \) lie.

To find between which two consecutive integers \( \sqrt{138} \) lies, we can start by estimating its square root. We know that \( 12^2 = 144 \) and \( 11^2 = 121 \), so \( \sqrt{138} \) is between \( 11 \) and \( 12 \). Specifically, \( 11.7^2 \) is approximately \( 136.89 \), which is less than \( 138 \), while \( 11.8^2 \) is approximately \( 139.24 \), which is greater than \( 138 \). Therefore, \( \sqrt{138} \) lies between \( 11 \) and \( 12 \). Now for \( q = \sqrt{b^2 - 4ac} \) with \( a = -1 \), \( b = -2 \), and \( c = 4 \), we first calculate the expression under the square root: \[ q = \sqrt{(-2)^2 - 4 \cdot (-1) \cdot 4} = \sqrt{4 + 16} = \sqrt{20} \] Next, simplifying \( \sqrt{20} \) gives us \( \sqrt{4 \times 5} = 2\sqrt{5} \). To find between which consecutive integers \( 2\sqrt{5} \) lies, we calculate \( \sqrt{5} \), which is about \( 2.236 \). Thus, \( 2\sqrt{5} \approx 4.472 \), indicating that \( q \) lies between \( 4 \) and \( 5 \).