For which value(s) of \( m \) will the following expressions have non-real roots? \text { (1) } x^{2}-6 x=m \text { (2) } x^{2}+m x+4=3 x \text { (3) } \frac{x^{2}-5 x+5}{2 x^{2}-x+1}=\frac{1}{m}

To determine the values of \( m \) for which the given expressions have non-real roots, we will analyze each expression separately. Non-real roots occur when the discriminant of a quadratic equation is less than zero. ### Expression (1): \( x^{2} - 6x = m \) Rearranging this gives us the standard form of a quadratic equation: \[ x^{2} - 6x - m = 0 \] The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ D = b^2 - 4ac \] For our equation, \( a = 1 \), \( b = -6 \), and \( c = -m \). Thus, the discriminant is: \[ D = (-6)^2 - 4(1)(-m) = 36 + 4m \] To find when the roots are non-real, we set the discriminant less than zero: \[ 36 + 4m < 0 \] Solving for \( m \): \[ 4m < -36 \\ m < -9 \] ### Expression (2): \( x^{2} + mx + 4 = 3x \) Rearranging gives: \[ x^{2} + mx - 3x + 4 = 0 \\ x^{2} + (m - 3)x + 4 = 0 \] Here, \( a = 1 \), \( b = m - 3 \), and \( c = 4 \). The discriminant is: \[ D = (m - 3)^2 - 4(1)(4) = (m - 3)^2 - 16 \] Setting the discriminant less than zero: \[ (m - 3)^2 - 16 < 0 \] This can be factored as: \[ (m - 3 - 4)(m - 3 + 4) < 0 \\ (m - 7)(m + 1) < 0 \] To find the intervals where this product is negative, we analyze the critical points \( m = 7 \) and \( m = -1 \). The intervals to test are \( (-\infty, -1) \), \( (-1, 7) \), and \( (7, \infty) \). - For \( m < -1 \): both factors are negative, product is positive. - For \( -1 < m < 7 \): one factor is negative, the other is positive, product is negative. - For \( m > 7 \): both factors are positive, product is positive. Thus, the roots are non-real for: \[ -1 < m < 7 \] ### Expression (3): \( \frac{x^{2} - 5x + 5}{2x^{2} - x + 1} = \frac{1}{m} \) Cross-multiplying gives: \[ m(x^{2} - 5x + 5) = 2x^{2} - x + 1 \] Rearranging leads to: \[ mx^{2} - 5mx + 5m - 2x^{2} + x - 1 = 0 \\ (m - 2)x^{2} + (-5m + 1)x + (5m - 1) = 0 \] Here, \( a = m - 2 \), \( b = -5m + 1 \), and \( c = 5m - 1 \). The discriminant is: \[ D = (-5m + 1)^2 - 4(m - 2)(5m - 1) \] Calculating the discriminant: \[ D = (25m^2 - 10m + 1) - 4[(m - 2)(5m - 1)] \] Expanding \( 4(m - 2)(5m - 1) \): \[ = 4(5m^2 - m - 10m + 2) = 20m^2 - 44m + 8 \] Thus, the discriminant becomes: \[ D = 25m^2 - 10m + 1 - (20m^2 - 44m + 8) \\ = 25m^2 - 10m + 1 - 20m^2 + 44m - 8 \\ = 5m^2 + 34m - 7 \] Setting this less than zero: \[ 5m^2 + 34m - 7 < 0 \] To find the roots of the quadratic \( 5m^2 + 34m - 7 = 0 \), we will calculate the discriminant: \[ D' = 34^2 - 4(5)(-7) = 1156 + 140 = 1296 \] Finding the roots: \[ m = \frac{-34 \pm \sqrt{1296}}{2 \cdot 5} = \frac{-34 \pm 36}{10} \] Calculating the roots: 1. \( m = \frac{2}{10} = 0.2 \) 2. \( m = \frac{-70}{10} = -7 \) The roots are \( m = -7 \) and \( m = 0.2 \). The quadratic opens upwards (since the coefficient of \( m^2 \) is positive), so it is negative between the roots: \[ -7 < m < 0.2 \] ### Summary of Results 1. For \( x^{2} - 6x = m \): \( m < -9 \) 2. For \( x^{2} + mx + 4 = 3x \): \( -1 < m < 7 \) 3. For \( \frac{x^{2} - 5x + 5}{2x^{2} - x + 1} = \frac{1}{m} \): \( -7 < m < 0.2 \) Thus, the values of \( m \) for which each expression has non-real roots are: 1. \( m < -9 \) 2. \( -1 < m < 7 \) 3. \( -7 < m < 0.2 \)