Solving Quadratic Equations with Complex Solutions - Algebra 1

1. Simplify $\sqrt{-75}$ in simplest form.

   $5\sqrt{3}i$. $\sqrt{-75} = \sqrt{25 \cdot 3} \cdot i = 5\sqrt{3}i$.

2. Simplify $\sqrt{-12}$ in simplest form.

   $2\sqrt{3}i$. $\sqrt{-12} = \sqrt{4 \cdot 3} \cdot i = 2\sqrt{3}i$.

3. Which condition on $\Delta$ guarantees two nonreal complex solutions?

   $\Delta<0$. Negative discriminant means the square root involves $\sqrt{\text{negative}}$.

4. Identify whether $x^2+6x+9=0$ has real solutions or complex solutions.

   Real solutions (a repeated real root). $\Delta = 36 - 36 = 0$, indicating one repeated real root.

5. Solve $(x+2)^2=-9$.

   $x=-2\pm 3i$. Take square root: $x + 2 = \pm\sqrt{-9} = \pm 3i$.

6. Solve $x^2-6x+13=0$.

   $x=3\pm 2i$. $\Delta = 36 - 52 = -16 < 0$, so solutions involve $i$.

7. Solve $4x^2+4x+5=0$.

   $x=-\frac{1}{2}\pm i$. $\Delta = 16 - 80 = -64 < 0$, then divide by $2a = 8$.

8. Solve $x^2+6x+34=0$.

   $x=-3\pm 5i$. $\Delta = 36 - 136 = -100 < 0$, yielding complex solutions.

9. Solve $x^2-18x+90=0$.

   $x=9\pm 3i$. $\Delta = 324 - 360 = -36 < 0$, so solutions are complex.

10. Solve $x^2+16=0$.

    $x=\pm 4i$. Rearrange to $x^2 = -16$, then $x = \pm\sqrt{-16} = \pm 4i$.

11. Solve $x^2+4x+5=0$.

    $x=-2\pm i$. $\Delta = 16 - 20 = -4 < 0$, so use quadratic formula with $i$.

12. Solve $x^2+14x+53=0$.

    $x=-7\pm 2i$. $\Delta = 196 - 212 = -16 < 0$, yielding complex solutions.

13. Solve $x^2= -49$.

    $x=\pm 7i$. Take square root of both sides: $x = \pm\sqrt{-49} = \pm 7i$.

14. Identify the sum of roots of $ax^2+bx+c=0$ in terms of $a$ and $b$.

    $r_1+r_2=-\frac{b}{a}$. Vieta's formula: sum of roots equals $-\frac{\text{coefficient of } x}{\text{leading coefficient}}$.

15. Find the discriminant of $x^2+4x+5=0$.

    $\Delta=-4$. $\Delta = b^2 - 4ac = 16 - 20 = -4$.

16. Identify the complex conjugate of $a+bi$.

    $a-bi$. Change the sign of the imaginary part.

17. Solve $x^2-12x+52=0$.

    $x=6\pm 4i$. $\Delta = 144 - 208 = -64 < 0$, so solutions are complex.

18. State the result of completing the square on $x^2+px$.

    $x^2+px=\left(x+\frac{p}{2}\right)^2-\left(\frac{p}{2}\right)^2$. Add and subtract $(\frac{p}{2})^2$ to create a perfect square trinomial.

19. What is the sum of the solutions to $x^2+4x+5=0$?

    $-4$. Sum of roots $= -\frac{b}{a} = -\frac{4}{1} = -4$.

20. State the vertex $x$-coordinate formula for $y=ax^2+bx+c$.

    $x=-\frac{b}{2a}$. Found by setting the derivative equal to zero or completing the square.

21. What is $\sqrt{-1}$ written as a complex unit?

    $i$. The imaginary unit, defined as $i^2 = -1$.

22. What is the discriminant of $ax^2+bx+c=0$?

    $\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.

23. Solve $x^2-4x+8=0$.

    $x=2\pm 2i$. $\Delta = 16 - 32 = -16 < 0$, so solutions are complex.

24. Identify the real part and imaginary part of the solution $3-2i$.

    Real part $3$, imaginary part $-2$. In $a + bi$ form, $a$ is real part and $b$ is imaginary part.

25. Find the discriminant of $3x^2-6x+5=0$.

    $\Delta=-24$. $\Delta = b^2 - 4ac = 36 - 60 = -24$.

26. What are the solutions of $x^2+1=0$?

    $x=\pm i$. Rearrange to $x^2 = -1$, then $x = \pm\sqrt{-1} = \pm i$.

27. Solve $x^2-8x+20=0$ by completing the square.

    $x=4\pm 2i$. Complete the square: $(x - 4)^2 = -4$, so $x - 4 = \pm 2i$.

28. Solve $2x^2-8x+17=0$.

    $x=2\pm \frac{\sqrt{2}}{2}i$. $\Delta = 64 - 136 = -72 < 0$, then divide by $2a = 4$.

29. Solve $x^2+18x+85=0$.

    $x=-9\pm 2i$. $\Delta = 324 - 340 = -16 < 0$, giving complex solutions.

30. What is the product of the solutions to $x^2+4x+5=0$?

    $5$. Product of roots $= \frac{c}{a} = \frac{5}{1} = 5$.

31. Solve $x^2+2x+5=0$ by completing the square.

    $x=-1\pm 2i$. Complete the square: $(x + 1)^2 = -4$, so $x + 1 = \pm 2i$.

32. What relationship do nonreal solutions have for a quadratic with real coefficients?

    They occur as conjugate pairs $a\pm bi$. Complex Conjugate Root Theorem for polynomials with real coefficients.

33. Solve $x^2-10x+29=0$.

    $x=5\pm 2i$. $\Delta = 100 - 116 = -16 < 0$, so solutions involve $i$.

34. Solve $x^2+2x+10=0$.

    $x=-1\pm 3i$. $\Delta = 4 - 40 = -36 < 0$, so solutions are complex.

35. Solve $x^2+8x+20=0$.

    $x=-4\pm 2i$. $\Delta = 64 - 80 = -16 < 0$, so solutions are nonreal.

36. Solve $x^2-14x+58=0$.

    $x=7\pm 3i$. $\Delta = 196 - 232 = -36 < 0$, so solutions involve $i$.

37. Solve $(2x-1)^2=-25$.

    $x=\frac{1}{2}\pm \frac{5}{2}i$. Take square root: $2x - 1 = \pm\sqrt{-25} = \pm 5i$.

38. Solve $3x^2+6x+7=0$.

    $x=-1\pm \frac{\sqrt{3}}{3}i$. $\Delta = 36 - 84 = -48 < 0$, then divide by $2a = 6$.

39. Solve $(x-3)^2=-16$.

    $x=3\pm 4i$. Take square root: $x - 3 = \pm\sqrt{-16} = \pm 4i$.

40. What is the standard simplification for $\sqrt{-k}$ when $k>0$?

    $\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the radical: $\sqrt{-k} = \sqrt{(-1) \cdot k}$.

41. State the quadratic formula for solutions of $ax^2+bx+c=0$.

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square or using algebraic manipulation.

42. Solve $5x^2+10x+13=0$.

    $x=-1\pm \frac{2}{5}i$. $\Delta = 100 - 260 = -160 < 0$, then divide by $2a = 10$.

43. Solve $x^2-2x+2=0$.

    $x=1\pm i$. $\Delta = 4 - 8 = -4 < 0$, giving complex solutions.

44. Simplify $\sqrt{-36}$.

    $6i$. $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$.

45. What are the solutions of $x^2+9=0$?

    $x=\pm 3i$. Rearrange to $x^2 = -9$, then $x = \pm\sqrt{-9} = \pm 3i$.

46. Simplify $\sqrt{-8}$ in simplest form.

    $2\sqrt{2}i$. $\sqrt{-8} = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2}i$.

47. Identify whether $x^2-2x+5=0$ has real solutions or complex solutions.

    Complex solutions (nonreal). $\Delta = 4 - 20 = -16 < 0$, indicating nonreal solutions.

48. Identify the product of roots of $ax^2+bx+c=0$ in terms of $a$ and $c$.

    $r_1r_2=\frac{c}{a}$. Vieta's formula: product of roots equals $\frac{\text{constant term}}{\text{leading coefficient}}$.

49. Solve $x^2+12x+40=0$.

    $x=-6\pm 2i$. $\Delta = 144 - 160 = -16 < 0$, giving complex solutions.

50. Solve $2x^2+4x+5=0$.

    $x=-1\pm \frac{\sqrt{6}}{2}i$. $\Delta = 16 - 40 = -24 < 0$, then divide by $2a = 4$.

51. Solve $x^2+10x+26=0$.

    $x=-5\pm i$. $\Delta = 100 - 104 = -4 < 0$, yielding complex solutions.

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