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

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UpStudy AI · Accepted Answer

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 $$x$$.
For $$a \neq b$$, $$x = -(a + b)$$. If $$a = b$$, the equation is true for any value of $$x$$.

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

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