\( a x-b^{2}=b x-a^{2}( \) in terms of \( a \) and \( b) \)

We start with the equation \[ a x - b^2 = b x - a^2. \] **Step 1.** Rearrange the equation by moving all terms with \(x\) to one side and constant terms to the other side: \[ a x - b x = b^2 - a^2. \] **Step 2.** Factor out \(x\) on the left-hand side: \[ (a - b)x = b^2 - a^2. \] **Step 3.** Factor the right-hand side as a difference of squares: \[ b^2 - a^2 = (b - a)(b + a). \] This gives: \[ (a - b)x = (b - a)(b + a). \] **Step 4.** Notice that \(b - a = -(a - b)\). Substitute this into the equation: \[ (a - b)x = -(a - b)(a + b). \] **Step 5.** Assuming \(a \neq b\) (so that \(a - b \neq 0\)), divide both sides by \(a - b\): \[ x = - (a + b). \] **Special Case:** If \(a = b\), then \(a - b = 0\) and the original equation becomes: \[ a x - a^2 = a x - a^2, \] which is true for all \(x\). **Final Answer:** - For \(a \neq b\), \[ x = - (a + b). \] - For \(a = b\), the equation holds for all \(x\).