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

To find the value(s) of \( m \) for which the given expressions have equal roots, we need to analyze each quadratic equation and apply the condition for equal roots, which is that the discriminant must be zero. ### Step 1: Analyze each equation 1. **Equation (1):** \( x^2 + 2m = 8x \) Rearranging gives: \[ x^2 - 8x + 2m = 0 \] The discriminant \( D_1 \) is given by: \[ D_1 = b^2 - 4ac = (-8)^2 - 4(1)(2m) = 64 - 8m \] For equal roots, set the discriminant to zero: \[ 64 - 8m = 0 \] Solving for \( m \): \[ 8m = 64 \implies m = 8 \] 2. **Equation (2):** \( x^2 + mx + 9 = 0 \) The discriminant \( D_2 \) is: \[ D_2 = m^2 - 4(1)(9) = m^2 - 36 \] For equal roots, set the discriminant to zero: \[ m^2 - 36 = 0 \] Solving for \( m \): \[ m^2 = 36 \implies m = 6 \text{ or } m = -6 \] 3. **Equation (3):** \( x^2 - mx + 2x + 1 = 0 \) Rearranging gives: \[ x^2 + (2 - m)x + 1 = 0 \] The discriminant \( D_3 \) is: \[ D_3 = (2 - m)^2 - 4(1)(1) = (2 - m)^2 - 4 \] For equal roots, set the discriminant to zero: \[ (2 - m)^2 - 4 = 0 \] Solving for \( m \): \[ (2 - m)^2 = 4 \implies 2 - m = 2 \text{ or } 2 - m = -2 \] This gives: \[ m = 0 \text{ or } m = 4 \] ### Step 2: Summary of results From the analysis, we have the following values of \( m \) for each equation to have equal roots: - From Equation (1): \( m = 8 \) - From Equation (2): \( m = 6 \) or \( m = -6 \) - From Equation (3): \( m = 0 \) or \( m = 4 \) ### Final Answer The values of \( m \) for which the expressions have equal roots are: \[ m = 8, \quad m = 6, \quad m = -6, \quad m = 0, \quad m = 4 \]