Solve Quadratics by Multiple Methods - Algebra 2

1. Factor and solve $x^2+7x+12=0$.

   $x=-3$ or $x=-4$. Factor: $(x+3)(x+4)=0$, then use zero-product property.

2. Solve using the quadratic formula: $x^2+4x+1=0$.

   $x=-2\pm\sqrt{3}$. Use $a=1$, $b=4$, $c=1$ in the quadratic formula.

3. Solve using the quadratic formula: $x^2+2x+5=0$.

   $x=-1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.

4. Solve by taking square roots: $(x+2)^2=7$.

   $x=-2\pm\sqrt{7}$. Square root both sides: $x+2=\pm\sqrt{7}$.

5. What is the zero-product property used after factoring $AB=0$?

   $AB=0\Rightarrow A=0$ or $B=0$. If a product equals zero, at least one factor must be zero.

6. What does $\Delta<0$ guarantee about solutions of $ax^2+bx+c=0$?

   Two complex conjugate solutions. Negative discriminant means the parabola doesn't cross the x-axis.

7. Solve using the quadratic formula: $2x^2+4x+5=0$.

   $x=-1\pm\frac{\sqrt{6}}{2}i$. Use $a=2$, $b=4$, $c=5$; $\sqrt{-24}=2\sqrt{6}i$.

8. Solve by factoring: $3x^2-12=0$.

   $x=\pm 2$. Factor: $3(x^2-4)=3(x-2)(x+2)=0$.

9. State the square root property used to solve $(x-h)^2=k$.

   $(x-h)^2=k\Rightarrow x-h=\pm\sqrt{k}$. Take the square root of both sides to isolate $x-h$.

10. What is $\Delta$ for $x^2+2x+5=0$?

    $\Delta=-16$. $\Delta=(2)^2-4(1)(5)=4-20=-16$.

11. Solve using the quadratic formula: $x^2+5x+6=0$.

    $x=-2$ or $x=-3$. Use $a=1$, $b=5$, $c=6$ in the quadratic formula.

12. Solve by completing the square: $x^2-4x-1=0$.

    $x=2\pm\sqrt{5}$. Complete the square: $(x-2)^2=5$, so $x-2=\pm\sqrt{5}$.

13. What are the solutions of $(x+5)^2=9$?

    $x=-2$ or $x=-8$. Square root both sides: $x+5=\pm 3$, so $x=-5\pm 3$.

14. Solve by completing the square: $x^2+6x+1=0$.

    $x=-3\pm 2\sqrt{2}$. Complete the square: $(x+3)^2=8$, so $x+3=\pm 2\sqrt{2}$.

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

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard form for finding roots of any quadratic equation.

16. What is the simplified form of $\sqrt{-12}$ in terms of $i$?

    $2\sqrt{3}i$. Simplify: $\sqrt{-12}=\sqrt{12}i=2\sqrt{3}i$.

17. Solve by factoring: $x^2+10x+25=0$.

    $x=-5$. Perfect square trinomial: $(x+5)^2=0$, so $x=-5$.

18. Solve using the quadratic formula: $3x^2-12x+12=0$.

    $x=2$. Use $a=3$, $b=-12$, $c=12$; discriminant equals 0.

19. Factor and solve $x^2-5x=0$.

    $x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$, then use zero-product property.

20. What is the simplified form of $\sqrt{-1}$?

    $i$. The imaginary unit where $i^2=-1$.

21. Factor and solve $x^2-9=0$.

    $x=\pm 3$. Difference of squares: $(x-3)(x+3)=0$.

22. What are the solutions of $x^2=49$?

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

23. Solve by taking square roots: $(2x-1)^2=25$.

    $x=3$ or $x=-2$. Square root both sides: $2x-1=\pm 5$, solve for $x$.

24. What does $\Delta>0$ guarantee about solutions of $ax^2+bx+c=0$?

    Two distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.

25. What is $\Delta$ for $2x^2-3x-2=0$?

    $\Delta=25$. $\Delta=(-3)^2-4(2)(-2)=9+16=25$.

26. What does $\Delta=0$ guarantee about solutions of $ax^2+bx+c=0$?

    One real double solution. Zero discriminant means the parabola touches the x-axis at one point.

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

    $x=2$ or $x=-4$. Complete the square: $(x+1)^2=9$, so $x+1=\pm 3$.

28. Factor and solve $x^2-6x+8=0$.

    $x=2$ or $x=4$. Factor: $(x-2)(x-4)=0$, then use zero-product property.

29. Identify the best method for $x^2-16=0$ from factoring or quadratic formula.

    Factoring. Difference of squares factors easily as $(x-4)(x+4)=0$.

30. What is the simplified form of $\sqrt{-49}$ in terms of $i$?

    $7i$. Use $\sqrt{-49}=\sqrt{49}\cdot\sqrt{-1}=7i$.

31. Identify $a$, $b$, and $c$ for $-3x^2+2x-7=0$.

    $a=-3$, $b=2$, $c=-7$. Coefficients from standard form $ax^2+bx+c=0$.

32. Solve by factoring: $x^2-2x-15=0$.

    $x=5$ or $x=-3$. Factor: $(x-5)(x+3)=0$, then use zero-product property.

33. What are the solutions of $(x-4)^2=0$?

    $x=4$. Only one solution since the square root of 0 is 0.

34. What is the discriminant for $ax^2+bx+c=0$?

    $\Delta=b^2-4ac$. The discriminant determines the nature of quadratic solutions.

35. What is $\Delta$ for $x^2-6x+9=0$?

    $\Delta=0$. $\Delta=(-6)^2-4(1)(9)=36-36=0$.

36. Solve by inspection: $x^2=\frac{1}{4}$.

    $x=\pm\frac{1}{2}$. Take the square root: $\sqrt{\frac{1}{4}}=\frac{1}{2}$.

37. Rewrite $x^2-12x+36$ as a perfect square trinomial.

    $(x-6)^2$. Perfect square trinomial with $a=x$ and $b=-6$.

38. How do you write solutions when the quadratic formula yields $\sqrt{-k}$ for $k>0$?

    Use $\sqrt{-k}=i\sqrt{k}$ to write $a\pm bi$. Convert negative square roots to imaginary form using $i$.

39. What number completes the square for $x^2-6x+\square$?

    $9$. Half of -6 is -3, and $(-3)^2=9$.

40. Identify the best method for $(x-9)^2=12$ from square roots or quadratic formula.

    Taking square roots. Already in perfect square form, ready for square root method.

41. What are the solutions of $(x-3)^2=16$?

    $x=7$ or $x=-1$. Square root both sides: $x-3=\pm 4$, so $x=3\pm 4$.

42. Solve by factoring: $x^2-4x+3=0$.

    $x=1$ or $x=3$. Factor: $(x-1)(x-3)=0$, then use zero-product property.

43. Rewrite $x^2+8x+16$ as a perfect square trinomial.

    $(x+4)^2$. Perfect square trinomial with $a=x$ and $b=4$.

44. Factor and solve $x^2+x-6=0$.

    $x=2$ or $x=-3$. Factor: $(x+3)(x-2)=0$, then use zero-product property.

45. Solve using the quadratic formula: $2x^2+3x-2=0$.

    $x=\frac{1}{2}$ or $x=-2$. Use $a=2$, $b=3$, $c=-2$ in the quadratic formula.

46. What are the solutions of $x^2+8x+16=0$?

    $x=-4$. Perfect square trinomial: $(x+4)^2=0$, so $x=-4$.

47. Solve using the quadratic formula: $x^2-2x+5=0$.

    $x=1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.

48. Solve by factoring: $2x^2-8x=0$.

    $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, then use zero-product property.

49. What is the simplified form of $\sqrt{-18}$ in terms of $i$?

    $3\sqrt{2}i$. Simplify: $\sqrt{-18}=\sqrt{18}i=3\sqrt{2}i$.

50. What number completes the square for $x^2+10x+\square$?

    $25$. Half of 10 is 5, and $5^2=25$.

51. What are the solutions of $2(x-1)^2=18$?

    $x=4$ or $x=-2$. Divide by 2 first: $(x-1)^2=9$, then $x-1=\pm 3$.

52. State the first step to complete the square for $x^2+bx$.

    Add $\left(\frac{b}{2}\right)^2$. Take half the coefficient of $x$, then square it.

Study these flashcards