1. **State the problem:** Solve the equation $$\sqrt{x} - 1 = x - 3$$ and determine which statements about the solutions are true. 2. **Isolate the square root:** Add 1 to both sides: $$\sqrt{x} = x - 2$$ 3. **Square both sides to eliminate the square root:** $$\left(\sqrt{x}\right)^2 = (x - 2)^2$$ $$x = (x - 2)^2$$ 4. **Expand the right side:** $$x = x^2 - 4x + 4$$ 5. **Bring all terms to one side:** $$0 = x^2 - 4x + 4 - x$$ $$0 = x^2 - 5x + 4$$ 6. **Factor the quadratic:** $$0 = (x - 4)(x - 1)$$ 7. **Find the roots:** $$x = 4 \quad \text{or} \quad x = 1$$ 8. **Check for extraneous solutions by substituting back into the original equation:** - For $$x=4$$: $$\sqrt{4} - 1 = 2 - 3$$ $$2 - 1 = -1$$ $$1 \neq -1$$ so $$x=4$$ is extraneous. - For $$x=1$$: $$\sqrt{1} - 1 = 1 - 3$$ $$1 - 1 = -2$$ $$0 \neq -2$$ so $$x=1$$ is extraneous. 9. **Check if any other values satisfy the original equation:** Recall from step 2: $$\sqrt{x} = x - 2$$ For $$x \geq 0$$, the right side must be non-negative: $$x - 2 \geq 0 \Rightarrow x \geq 2$$ Try $$x=5$$: $$\sqrt{5} - 1 \approx 2.236 - 1 = 1.236$$ $$5 - 3 = 2$$ Not equal. Try $$x=2$$: $$\sqrt{2} - 1 \approx 1.414 - 1 = 0.414$$ $$2 - 3 = -1$$ Not equal. 10. **Conclusion:** No solutions satisfy the original equation. Both roots from the squared equation are extraneous. **Answer:** - A. 5 and 2 are extraneous solutions. **False** (5 and 2 are not solutions). - B. The solution of the equation is 5. **False**. - C. The equation has only one solution. **False**. - D. The solution set of the equation is {5, 2}. **False**. **Final note:** The equation has no real solutions.