1. **Problem:** Given the equations $$a = \frac{1 - x^2}{1 + x^2}$$ and $$b = 4 - 2x,$$ express $b$ in terms of $a$. 2. **Formula and rules:** We want to find $b$ as a function of $a$, so we need to express $x$ from the first equation and substitute it into the second. 3. **Step 1: Express $x^2$ from $a$:** Start with $$a = \frac{1 - x^2}{1 + x^2}$$ Multiply both sides by $1 + x^2$: $$a(1 + x^2) = 1 - x^2$$ Expand: $$a + a x^2 = 1 - x^2$$ Bring terms with $x^2$ to one side: $$a x^2 + x^2 = 1 - a$$ Factor $x^2$: $$x^2(a + 1) = 1 - a$$ Solve for $x^2$: $$x^2 = \frac{1 - a}{a + 1}$$ 4. **Step 2: Express $x$:** Since $x^2 = \frac{1 - a}{a + 1}$, then $$x = \pm \sqrt{\frac{1 - a}{a + 1}}$$ 5. **Step 3: Substitute $x$ into $b$:** Recall $$b = 4 - 2x$$ Substitute $x$: $$b = 4 - 2 \left( \pm \sqrt{\frac{1 - a}{a + 1}} \right)$$ 6. **Final expression:** $$b = 4 \mp 2 \sqrt{\frac{1 - a}{a + 1}}$$ This expresses $b$ in terms of $a$ with two possible values depending on the sign of $x$. **Answer:** $$b = 4 \pm 2 \sqrt{\frac{1 - a}{a + 1}}$$