x2-4x+3 =0 2x2 + 5x - 3 = 0 3x2 - 12 = 0 x2 + 6x + 9 = 0 x2 -2x - 8=0 4x2 + x - 10 = 0 5x2 - 20x + 15 = 0 x* + 4x = 0 9x2 - 24x + 16 = 0 2x2 - 3x + 1 = 0 x₴ - 6 = 0 x2 + 2x+1=0…

Let's solve each quadratic equation step-by-step.

1. \( x^2 - 4x + 3 = 0 \)
   Factoring: \( (x-1)(x-3) = 0 \)
   Solutions: \( x = 1, 3 \)

2. \( 2x^2 + 5x - 3 = 0 \)
   Using the quadratic formula:
   \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)
   Here, \( a = 2, b = 5, c = -3 \)
   \( x = \frac{{-5 \pm \sqrt{{5^2 - 4 \cdot 2 \cdot (-3)}}}}{2 \cdot 2} \)
   \( x = \frac{{-5 \pm \sqrt{{49}}}}{4} \)
   \( x = \frac{{-5 \pm 7}}{4} \)
   Solutions: \( x = \frac{1}{2}, -3 \)

3. \( 3x^2 - 12 = 0 \)
   Simplifying:
   \( 3x^2 = 12 \)
   \( x^2 = 4 \)
   Solutions: \( x = 2, -2 \)

4. \( x^2 + 6x + 9 = 0 \)
   Factoring: \( (x+3)^2 = 0 \)
   Solution: \( x = -3 \) (double root)

5. \( x^2 - 2x - 8 = 0 \)
   Factoring: \( (x-4)(x+2) = 0 \)
   Solutions: \( x = 4, -2 \)

6. \( 4x^2 + x - 10 = 0 \)
   Using the quadratic formula:
   Here, \( a = 4, b = 1, c = -10 \)
   \( x = \frac{{-1 \pm \sqrt{{1 + 160}}}}{8} \)
   \( x = \frac{{-1 \pm 13}}{8} \)
   Solutions: \( x = \frac{3}{4}, -\frac{7}{4} \)

7. \( 5x^2 - 20x + 15 = 0 \)
   Dividing through by 5:
   \( x^2 - 4x + 3 = 0 \) (same as problem 1)
   Solutions: \( x = 1, 3 \)

8. \( x^2 + 4x = 0 \)
   Factoring: \( x(x + 4) = 0 \)
   Solutions: \( x = 0, -4 \)

9. \( 9x^2 - 24x + 16 = 0 \)
   Using the quadratic formula:
   Here, \( a = 9, b = -24, c = 16 \)
   \( x = \frac{{24 \pm \sqrt{{576 - 576}}}}{2 \cdot 9} \)
   Solution: \( x = \frac{24}{18} = \frac{4}{3} \) (double root)

10. \( 2x^2 - 3x + 1 = 0 \)
    Using the quadratic formula:
    \( a = 2, b = -3, c = 1 \)
    \( x = \frac{{3 \pm \sqrt{{-1}}}}{4} \)
    Solutions: \( x = \frac{3}{4} \pm \frac{i}{2} \)

11. \( x^2 - 6 = 0 \)
    Rearranging:
    \( x^2 = 6 \)
    Solutions: \( x = \sqrt{6}, -\sqrt{6} \)

12. \( x^2 + 2x + 1 = 0 \)
    Factoring: \( (x+1)^2 = 0 \)
    Solution: \( x = -1 \) (double root)

13. \( 7x^2 + 2x - 14 = 0 \)
    Using the quadratic formula:
    \( a = 7, b = 2, c = -14 \)
    \( x = \frac{{-2 \pm \sqrt{{4 + 392}}}}{14} \)
    \( x = \frac{{-2 \pm 20}}{14} \)
    Solutions: \( x = \frac{9}{7}, -\frac{11}{7} \)

14. \( x^2 - 10x + 25 = 0 \)
    Factoring: \( (x-5)^2 = 0 \)
    Solution: \( x = 5 \) (double root)

15. \( 6x^2 + 5 = 0 \)
    Rearranging:
    \( 6x^2 = -5 \)
    Solutions: \( x = \pm \sqrt{-\frac{5}{6}} = \pm \frac{i\sqrt{30}}{6} \)

16. \( x^2 + 5x - 14 = 0 \)
    Using the quadratic formula:
    \( a = 1, b = 5, c = -14 \)
    \( x = \frac{{-5 \pm \sqrt{{25 + 56}}}}{2} \)
    \( x = \frac{{-5 \pm 11}}{2} \)
    Solutions: \( x = 3, -8 \)

17. \( 8x - 2x + 1 = 0 \)
    Simplifying:
    \( 6x + 1 = 0 \)
    Solution: \( x = -\frac{1}{6} \)

18. \( 3x^2 + 12x = 0 \)
    Factoring: \( 3x(x + 4) = 0 \)
    Solutions: \( x = 0, -4 \)

19. \( x^2 - 3x - 4 = 0 \)
    Factoring: \( (x - 4)(x + 1) = 0 \)
    Solutions: \( x = 4, -1 \)

20. \( 10x^2 + 1 - 20 = 0 \)
    Rearranging:
    \( 10x^2 - 19 = 0 \)
    Solutions: \( x = \pm \sqrt{\frac{19}{10}} \)

21. \( x^2 + 8 = 0 \)
    Rearranging:
    \( x^2 = -8 \)
    Solutions: \( x = \pm 2i\sqrt{2} \)

22. \( 2x^2 - 8x + 6 = 0 \)
    Using the quadratic formula:
    \( a = 2, b = -8, c = 6 \)
    \( x = \frac{{8 \pm \sqrt{{64 - 48}}}}{4} \)
    Solutions: \( x = 3, 1.5 \)

23. \( 4x^2 + 9x + 2 = 0 \)
    Using the quadratic formula:
    \( a = 4, b = 9, c = 2 \)
    \( x = \frac{{-9 \pm \sqrt{{81 - 32}}}}{8} \)
    Solutions: \( x = \frac{-9 \pm \sqrt{49}}{8} = \frac{-9 \pm 7}{8} \)
    Solutions: \( x = -\frac{1}{4}, -2 \)

24. \( 5x^2 - 5x = 0 \)
    Factoring: \( 5x(x - 1) = 0 \)
    Solutions: \( x = 0, 1 \)

25. \( x^2 + x - 6 = 0 \)
    Factoring: \( (x - 2)(x + 3) = 0 \)
    Solutions: \( x = 2, -3 \)

26. \( 3x - 27 = 0 \)
    Rearranging:
    \( 3x = 27 \)
    Solution: \( x = 9 \)

27. \( 2x^2 + 3x + 1 = 0 \)
    Using the quadratic formula:
    \( a = 2, b = 3, c = 1 \)
    \( x = \frac{{-3 \pm \sqrt{9 - 8}}}{4} \)
    Solutions: \( x = -1, -\frac{1}{2} \)

28. \( x^2 - 8x + 16 = 0 \)
    Factoring: \( (x - 4)^2 = 0 \)
    Solution: \( x = 4 \) (double root)

29. \( x^2 + 10x + 25 = 0 \)
    Factoring: \( (x + 5)^2 = 0 \)
    Solution: \( x = -5 \) (double root)

30. \( 6x^2 - 12x = 0 \)
    Factoring: \( 6x(x - 2) = 0 \)
    Solutions: \( x = 0, 2 \)

This completes the solutions for the equations provided.