1. **State the problem:** Solve for $x$ in the equation $$\sqrt{14.44} + \sqrt{9 + x^2} = 8.8.$$\n\n2. **Calculate the known square root:** \n$$\sqrt{14.44} = 3.8.$$\n\n3. **Rewrite the equation:** \n$$3.8 + \sqrt{9 + x^2} = 8.8.$$\n\n4. **Isolate the square root term:** \n$$\sqrt{9 + x^2} = 8.8 - 3.8 = 5.$$\n\n5. **Square both sides to eliminate the square root:** \n$$\left(\sqrt{9 + x^2}\right)^2 = 5^2$$\n$$9 + x^2 = 25.$$\n\n6. **Solve for $x^2$:** \n$$x^2 = 25 - 9 = 16.$$\n\n7. **Find $x$ by taking the square root:** \n$$x = \pm \sqrt{16} = \pm 4.$$\n\n**Final answer:** $$x = 4 \text{ or } x = -4.$$