1. **Problem a:** Solve the equation $$\sqrt{4x - 7} + 2 = 5$$ 2. **Step 1:** Isolate the square root term: $$\sqrt{4x - 7} = 5 - 2 = 3$$ 3. **Step 2:** Square both sides to eliminate the square root: $$4x - 7 = 3^2 = 9$$ 4. **Step 3:** Solve for $x$: $$4x = 9 + 7 = 16$$ $$x = \frac{16}{4} = 4$$ 5. **Step 4:** Check the solution in the original equation to avoid extraneous roots: $$\sqrt{4(4) - 7} + 2 = \sqrt{16 - 7} + 2 = \sqrt{9} + 2 = 3 + 2 = 5$$ Solution $x=4$ is valid. --- 6. **Problem b:** Solve the quadratic equation $$-x^2 + 2x - 2 = 0$$ using the quadratic formula. 7. **Step 1:** Write the quadratic in standard form: $$-x^2 + 2x - 2 = 0$$ Coefficients: $a = -1$, $b = 2$, $c = -2$ 8. **Step 2:** Quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 9. **Step 3:** Calculate the discriminant: $$\Delta = b^2 - 4ac = 2^2 - 4(-1)(-2) = 4 - 8 = -4$$ 10. **Step 4:** Since $\Delta < 0$, there are no real solutions; solutions are complex: $$x = \frac{-2 \pm \sqrt{-4}}{2(-1)} = \frac{-2 \pm 2i}{-2} = 1 \mp i$$ --- 11. **Problem 4a:** Write the equation of the circle $$x^2 + y^2 + 10x - 14y + 21 = 0$$ in standard form and identify center and radius. 12. **Step 1:** Group $x$ and $y$ terms: $$ (x^2 + 10x) + (y^2 - 14y) = -21 $$ 13. **Step 2:** Complete the square for $x$ and $y$: - For $x$: half of 10 is 5, square is 25 - For $y$: half of -14 is -7, square is 49 Add 25 and 49 to both sides: $$ (x^2 + 10x + 25) + (y^2 - 14y + 49) = -21 + 25 + 49 $$ 14. **Step 3:** Write as squares: $$ (x + 5)^2 + (y - 7)^2 = 53 $$ 15. **Step 4:** Identify center and radius: Center: $(-5, 7)$ Radius: $\sqrt{53}$ --- 16. **Problem 4b:** Determine the vertex and axis of symmetry of the parabola defined by $$f(x) = 2x^2 - 8x + 6$$ 17. **Step 1:** Use vertex formula: $$x_{vertex} = -\frac{b}{2a} = -\frac{-8}{2 \times 2} = \frac{8}{4} = 2$$ 18. **Step 2:** Find $y$ coordinate of vertex: $$f(2) = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2$$ 19. **Step 3:** Vertex is at $(2, -2)$ 20. **Step 4:** Axis of symmetry is the vertical line through the vertex: $$x = 2$$