For which value(s) of \( m \) will the following expressions have equal roots? \( \begin{array}{ll}\text { (1) } x^{2}+2 m=8 x & \text { (2) } x^{2}+m x+9=0 \\ \text { (3) } x^{2}-m x+2 x+1=0 & \text { (2) }\end{array} \)

To find the value(s) of \( m \) for which the given expressions have equal roots, we need to analyze the discriminant of each quadratic equation. 1. For the first equation \( x^2 - 8x + 2m = 0 \): The discriminant \( D_1 \) is given by: \[ D_1 = b^2 - 4ac = (-8)^2 - 4(1)(2m) = 64 - 8m \] For the roots to be equal, the discriminant must be zero: \[ 64 - 8m = 0 \implies 8m = 64 \implies m = 8 \] 2. For the second equation \( x^2 + mx + 9 = 0 \): The discriminant \( D_2 \) is: \[ D_2 = m^2 - 4(1)(9) = m^2 - 36 \] Setting the discriminant equal to zero gives: \[ m^2 - 36 = 0 \implies m^2 = 36 \implies m = 6 \text{ or } m = -6 \] 3. For the third equation \( x^2 + (2-m)x + 1 = 0 \): The discriminant \( D_3 \) is: \[ D_3 = (2-m)^2 - 4(1)(1) = (2-m)^2 - 4 \] Setting this discriminant to zero: \[ (2-m)^2 - 4 = 0 \implies (2-m)^2 = 4 \implies 2-m = 2 \text{ or } 2-m = -2 \] From \( 2-m=2 \): \[ m=0 \] From \( 2-m=-2 \): \[ m=4 \] Now, summarizing the values of \( m \): - From equation (1): \( m = 8 \) - From equation (2): \( m = 6 \) or \( m = -6 \) - From equation (3): \( m = 0 \) or \( m = 4 \) Thus, the values of \( m \) for which at least one expression has equal roots are \( m = 8, 6, -6, 0, 4 \).