1. **Stating the problem:** We want to understand and solve the equation $$\frac{\sqrt{2}}{2} = \frac{2x + 9}{\sqrt{x^2 + 74.3}}$$ and the equivalent transformations given. 2. **Original equation:** $$\frac{\sqrt{2}}{2} = \frac{2x + 9}{\sqrt{x^2 + 74.3}}$$ 3. **Goal:** Solve for $x$ by isolating it. 4. **Step 1: Cross-multiply to eliminate the fraction:** $$\sqrt{2} \cdot \sqrt{x^2 + 74.3} = 2 \cdot (2x + 9)$$ 5. **Simplify the right side:** $$\sqrt{2} \cdot \sqrt{x^2 + 74.3} = 4x + 18$$ 6. **Step 2: Use the property of square roots:** $$\sqrt{2} \cdot \sqrt{x^2 + 74.3} = \sqrt{2(x^2 + 74.3)} = \sqrt{2x^2 + 148.6}$$ 7. **Rewrite the equation:** $$\sqrt{2x^2 + 148.6} = 4x + 18$$ 8. **Step 3: Square both sides to remove the square root:** $$\left(\sqrt{2x^2 + 148.6}\right)^2 = (4x + 18)^2$$ $$2x^2 + 148.6 = (4x + 18)^2$$ 9. **Expand the right side:** $$(4x + 18)^2 = 16x^2 + 144x + 324$$ 10. **Set up the quadratic equation:** $$2x^2 + 148.6 = 16x^2 + 144x + 324$$ 11. **Bring all terms to one side:** $$0 = 16x^2 + 144x + 324 - 2x^2 - 148.6$$ $$0 = 14x^2 + 144x + 175.4$$ 12. **Simplify:** $$14x^2 + 144x + 175.4 = 0$$ 13. **Step 4: Solve the quadratic equation using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=14$, $b=144$, $c=175.4$. 14. **Calculate the discriminant:** $$\Delta = 144^2 - 4 \cdot 14 \cdot 175.4 = 20736 - 9822.4 = 10913.6$$ 15. **Calculate the roots:** $$x = \frac{-144 \pm \sqrt{10913.6}}{28}$$ $$\sqrt{10913.6} \approx 104.47$$ 16. **Two solutions:** $$x_1 = \frac{-144 + 104.47}{28} = \frac{-39.53}{28} \approx -1.41$$ $$x_2 = \frac{-144 - 104.47}{28} = \frac{-248.47}{28} \approx -8.87$$ 17. **Step 5: Check for extraneous solutions by substituting back into the original equation (optional but recommended).** **Final answer:** $$x \approx -1.41 \text{ or } x \approx -8.87$$