Study Guide Unit 7B

Solving Quadratic Equations using the Quadratic Formula

1. Equation: (x^2 + 2x - 3 = 0)

   • Solutions: (x = 1), (x = -3)

2. Equation: (3x^2 + 5x + 1 = 0)

   • Solutions:
     • (x \approx 0.23)
     • (x \approx 1.43)

3. Equation: (2x^2 - 4x + 5 = 0)

   • Solutions:
     • (x \approx 0.87)
     • (x \approx 2.87)

4. Equation: (5x^2 + 6x - 5 = 0)

   • Quadratic formula setup is complex, detailed steps omitted.

Other Methods to Solve Quadratics

5. Equation: (3x^2 - 5x - 12 = 0)

   • Solution requires alternative methods not detailed here.

6. Equation: (4(x - 3)^2 + 13 = 209)

   • Solution requires alternative methods not detailed here.

7. Equation: (12x^2 + (2x + 1)^2 = 372)

   • Solution requires alternative methods not detailed here.

Simplifying Square Root Expressions

8. (\sqrt{68})
9. (\sqrt{396})
10. (\sqrt{9136})
11. (\sqrt{24})
12. (\frac{-37}{\sqrt{5}})
13. (\frac{2}{\sqrt{65}})

Imaginary and Complex Numbers

14. Simplify: (\sqrt{-81})
15. Simplify: (\sqrt{-625})
16. Expression: ((9i)^2)
17. Expression: ((3 + 2i) + (7 + 6i))
18. Expression: ((10 + 5i) - (10 - 8i))
19. Expression: ((3 + 2i)(4 + 5i))
20. Expression: ((4 + 5i)^2)
21. Expression: ((7 - 2i)^2)

Additional Problems

22. Equation: (x^2 - 4x + 10)
    • Additional manipulation or solving required.
23. Expression to Simplify: (8i^2 + 15i)
24. Expression to Simplify: (24 - 213i - 5i^2)

Notes:

• The lecture focuses on solving quadratic equations using various methods, including the quadratic formula.
• Simplifying square roots and working with imaginary and complex numbers is emphasized.
• Some problems were noted with missing steps, indicating alternative methods need to be applied.
• The content also includes specific examples of complex number operations.