\( 2 x^{2}=-x+5( \) Are the roots real or Unreal

It appears there might be a small typo in your equation. Assuming the equation is: \[ 2x^{2} = -x + 5 \] Let's analyze the roots of the quadratic equation. ### Step 1: Rewrite the Equation in Standard Form First, bring all terms to one side to express the equation in the standard quadratic form \( ax^2 + bx + c = 0 \): \[ 2x^2 + x - 5 = 0 \] ### Step 2: Calculate the Discriminant The discriminant (\( D \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ D = b^2 - 4ac \] Substituting the values from the equation: - \( a = 2 \) - \( b = 1 \) - \( c = -5 \) \[ D = (1)^2 - 4(2)(-5) \] \[ D = 1 + 40 \] \[ D = 41 \] ### Step 3: Determine the Nature of the Roots - **If \( D > 0 \)**: There are two distinct real roots. - **If \( D = 0 \)**: There is exactly one real root (a repeated root). - **If \( D < 0 \)**: The roots are complex (unreal). Since \( D = 41 > 0 \), the quadratic equation has **two distinct real roots**. ### Conclusion The roots of the equation \( 2x^{2} = -x + 5 \) are **real and distinct**.