Complete the Square to Find Solutions - Algebra 2

1. Identify the missing term: $x^2-3x+\_\_\_ = \left(x-\frac{3}{2}\right)^2$.

   $\frac{9}{4}$. $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$ completes the square.

2. Rewrite $x^2-5x$ as a square plus a constant: $x^2-5x=\left(x-\_\_\right)^2-\_\_$.

   $\left(x-\frac{5}{2}\right)^2-\frac{25}{4}$. Half of $ -5 $ is $-\frac{5}{2}$, then subtract $\left(\frac{5}{2}\right)^2$.

3. What is the result after dividing $ax^2+bx+c=0$ by $a$?

   $x^2+\frac{b}{a}x+\frac{c}{a}=0$. Standard form after dividing by the leading coefficient.

4. What does $b^2-4ac<0$ tell you about the solutions of $ax^2+bx+c=0$?

   No real solutions (two complex solutions). Negative discriminant means no real intersection points.

5. Solve by completing the square: $4x^2+4x-3=0$.

   $x=\frac{1}{2}$ or $x=-\frac{3}{2}$. From $(x+\frac{1}{2})^2=1$, solve: $x=-\frac{1}{2}\pm 1$.

6. What is the quadratic formula for solutions to $ax^2+bx+c=0$?

   $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general form.

7. What is the goal form when completing the square for a quadratic in $x$?

   $(x-p)^2 = q$. This standard form isolates the squared term and constant.

8. What is $h$ in vertex form $y=a(x-h)^2+k$ in terms of $a$ and $b$?

   $h=-\frac{b}{2a}$. Formula for the x-coordinate of the vertex.

9. What perfect square trinomial matches $x^2-8x+16$?

   $\left(x-4\right)^2$. Perfect square with $p=4$ from $-8x$ coefficient.

10. After completing the square, what equation do you get for $ax^2+bx+c=0$ before square-rooting?

    $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$. The completed square form before taking square roots.

11. Transform $x^2-6x+9=0$ into $(x-p)^2=q$ form.

    $\left(x-3\right)^2=0$. This is already a perfect square trinomial.

12. What is the completing-the-square step after getting $x^2+bx$ on one side?

    Add $\left(\frac{b}{2}\right)^2$ to both sides. Maintains equation balance while creating a perfect square.

13. Solve by completing the square: $x^2-4x-5=0$.

    $x=5$ or $x=-1$. From $(x-2)^2=9$, take square root: $x=2\pm 3$.

14. Solve by completing the square: $2x^2+8x+3=0$.

    $x=-2\pm\frac{\sqrt{10}}{2}$. From $(x+2)^2=\frac{5}{2}$, solve: $x=-2\pm\sqrt{\frac{5}{2}}$.

15. Solve mentally: $\left(x-2\right)^2=16$.

    $x=6$ or $x=-2$. Square root of $16$ is $4$, so $x-2=\pm 4$.

16. What are the solutions of $(x-p)^2=q$ written explicitly?

    $x=p\pm\sqrt{q}$. Explicit form after isolating $x$.

17. Identify the value added to complete the square in $x^2-10x+\_\_$.

    $25$. Half of $-10$ is $-5$, squared gives $25$.

18. Solve mentally: $\left(x+5\right)^2=1$.

    $x=-4$ or $x=-6$. Square root of $1$ is $1$, so $x+5=\pm 1$.

19. Identify the value of $h$ (axis of symmetry) for $ax^2+bx+c$ using completing-the-square facts.

    $h=-\frac{b}{2a}$. Vertex x-coordinate from completing the square method.

20. What does $b^2-4ac>0$ tell you about the solutions of $ax^2+bx+c=0$?

    Two distinct real solutions. Positive discriminant means two real intersection points.

21. How many real solutions does $x^2-12x+40=0$ have after writing $\left(x-6\right)^2=-4$?

    Zero real solutions. Negative right side means no real square roots exist.

22. What operation do you use after writing a quadratic as $(x-p)^2=q$ to solve for $x$?

    Take square roots: $x-p=\pm\sqrt{q}$. Square root both sides to solve for $x$.

23. Solve mentally: $\left(x+3\right)^2=0$.

    $x=-3$. When the square equals zero, there's one solution.

24. Identify $p$ and $q$ if $\left(x-4\right)^2=9$ is in $(x-p)^2=q$ form.

    $p=4,\ q=9$. Standard $(x-p)^2=q$ form identification.

25. Identify the common error: completing the square for $x^2+10x$ by adding $10^2$; what should be added?

    $\left(\frac{10}{2}\right)^2=25$. The correct term is $\left(\frac{10}{2}\right)^2$, not $(10)^2$.

26. What is the next step after $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$?

    $x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}$. Take square root of both sides to solve.

27. What do you get after isolating $x$ from $x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. The quadratic formula derived from completing the square.

28. What expression is the discriminant in $ax^2+bx+c=0$?

    $b^2-4ac$. Determines the nature of quadratic solutions.

29. Transform $x^2+6x+1=0$ into $(x-p)^2=q$ form.

    $\left(x+3\right)^2=8$. Complete square: add $9$ to both sides, then rearrange.

30. What is the first step to complete the square in $ax^2+bx+c=0$ when $a\neq 1$?

    Divide by $a$ to make the $x^2$ coefficient $1$. Makes the leading coefficient 1 for easier completion.

31. Solve mentally: $\left(x-\frac{1}{2}\right)^2=\frac{9}{4}$.

    $x=2$ or $x=-1$. Square root of $\frac{9}{4}$ is $\frac{3}{2}$, so solutions differ by $3$.

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

    $x=-3\pm 2\sqrt{2}$. From $(x+3)^2=8$, take square root: $x=-3\pm 2\sqrt{2}$.

33. What is the key identity used to expand $(x+p)^2$ while completing the square?

    $(x+p)^2=x^2+2px+p^2$. Fundamental binomial expansion used in completing squares.

34. What value is added to $x^2+bx$ to complete the square?

    $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.

35. Simplify $-\frac{c}{a}+\left(\frac{b}{2a}\right)^2$ as a single fraction.

    $\frac{b^2-4ac}{4a^2}$. Common denominator simplification of the right side.

36. Solve by completing the square: $3x^2-12x+1=0$.

    $x=2\pm\frac{\sqrt{33}}{3}$. From $(x-2)^2=\frac{11}{3}$, solve: $x=2\pm\sqrt{\frac{11}{3}}$.

37. What square expression forms from $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2$?

    $\left(x+\frac{b}{2a}\right)^2$. The perfect square trinomial after completing.

38. Solve by completing the square: $x^2+2x-7=0$.

    $x=-1\pm 2\sqrt{2}$. From $(x+1)^2=8$, take square root: $x=-1\pm 2\sqrt{2}$.

39. What perfect square trinomial equals $x^2+bx+\left(\frac{b}{2}\right)^2$?

    $\left(x+\frac{b}{2}\right)^2$. The completed perfect square trinomial form.

40. Solve by completing the square: $x^2-6x+9=0$.

    $x=3$. Perfect square equals zero gives one repeated root.

41. What is the vertex form obtained by completing the square for $y=ax^2+bx+c$?

    $y=a(x-h)^2+k$. Standard vertex form from completing the square.

42. Transform $2x^2+8x+3=0$ into $(x-p)^2=q$ form.

    $\left(x+2\right)^2=\frac{5}{2}$. First divide by $2$, then complete the square.

43. What does $b^2-4ac=0$ tell you about the solutions of $ax^2+bx+c=0$?

    One real double root. Zero discriminant means one repeated real solution.

44. Transform $4x^2+4x-3=0$ into $(x-p)^2=q$ form.

    $\left(x+\frac{1}{2}\right)^2=1$. First divide by $4$, then complete the square.

45. After $x^2+\frac{b}{a}x=-\frac{c}{a}$, what is added to both sides to complete the square?

    $\left(\frac{b}{2a}\right)^2$. Half the new coefficient of $x$, then squared.

46. Transform $x^2-4x-5=0$ into $(x-p)^2=q$ form.

    $\left(x-2\right)^2=9$. Complete square: add $4$ to both sides, then rearrange.

47. Rewrite $x^2+9x$ as a square plus a constant: $x^2+9x=\left(x+\_\_\right)^2-\_\_$.

    $\left(x+\frac{9}{2}\right)^2-\frac{81}{4}$. Half of $9$ is $\frac{9}{2}$, then subtract $\left(\frac{9}{2}\right)^2$.

48. What perfect square trinomial matches $x^2+12x+36$?

    $\left(x+6\right)^2$. Perfect square with $p=-6$ from $+12x$ coefficient.

49. Transform $x^2+2x-7=0$ into $(x-p)^2=q$ form.

    $\left(x+1\right)^2=8$. Complete square: add $1$ to both sides, then rearrange.

50. How many real solutions does $x^2+4x+10=0$ have after writing $\left(x+2\right)^2=-6$?

    Zero real solutions. Negative right side means no real square roots exist.

51. What does the symbol $\pm$ indicate when solving $(x-p)^2=q$?

    Two cases: $+$ and $-$ square roots. Plus-minus accounts for both positive and negative square roots.

52. Identify the value added to complete the square in $x^2+7x+\_\_$.

    $\frac{49}{4}$. Half of $7$ is $\frac{7}{2}$, squared gives $\frac{49}{4}$.

53. Transform $3x^2-12x+1=0$ into $(x-p)^2=q$ form.

    $\left(x-2\right)^2=\frac{11}{3}$. First divide by $3$, then complete the square.

54. Identify $p$ and $q$ if $\left(x+\frac{1}{3}\right)^2=\frac{7}{9}$ is in $(x-p)^2=q$ form.

    $p=-\frac{1}{3},\ q=\frac{7}{9}$. Note: $(x+\frac{1}{3})^2 = (x-(-\frac{1}{3}))^2$.

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