Complete the Square to Find Solutions - Algebra 1

1. What is $p$ and $q$ if $x^2-6x+1=0$ is written as $\left(x-p\right)^2=q$?

   $p=3,\ q=8$. From $x^2-6x+1=0$: $(x-3)^2=9-1=8$.

2. What operation isolates $x$ after obtaining $\left(x-p\right)^2=q$?

   Take square roots: $x-p=\pm\sqrt{q}$. Taking square roots of both sides gives the plus-minus solutions.

3. What are the solutions of $x^2+6x+5=0$?

   $x=-1$ or $x=-5$. From $(x+3)^2=4$: $x+3=\pm 2$, so $x=-3\pm 2$.

4. What are the solutions of $x^2+4x-1=0$?

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

5. What are the solutions of $5x^2+20x+15=0$?

   $x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.

6. What are the solutions of $2x^2-4x-1=0$?

   $x=1\pm\frac{\sqrt{6}}{2}$. From $(x-1)^2=\frac{3}{2}$: $x=1\pm\sqrt{\frac{3}{2}}=1\pm\frac{\sqrt{6}}{2}$.

7. What is the completed-square form of $3x^2-12x+9=0$?

   $\left(x-2\right)^2=1$. First divide by 3, then complete the square on $x^2-4x+3=0$.

8. What are the solutions of $\left(x-\frac{1}{2}\right)^2=\frac{25}{4}$?

   $x=3$ or $x=-2$. From $x-\frac{1}{2}=\pm\frac{5}{2}$: $x=\frac{1}{2}\pm\frac{5}{2}$.

9. What is the completed-square form of $x^2+4x-1=0$?

   $\left(x+2\right)^2=5$. Adding $\left(\frac{4}{2}\right)^2=4$ to both sides, then simplifying.

10. What property justifies adding the same number to both sides while completing the square?

    Addition Property of Equality. This allows adding the same value to both sides of an equation.

11. What are the solutions of $x^2+6x+13=0$?

    $x=-3\pm 2i$. From $(x+3)^2=-4$: $x=-3\pm\sqrt{-4}=-3\pm 2i$.

12. What are the solutions of $x^2-2x-8=0$?

    $x=4$ or $x=-2$. From $(x-1)^2=9$: $x-1=\pm 3$, so $x=1\pm 3$.

13. What conclusion about real solutions follows from $\left(x+1\right)^2=-9$?

    No real solutions. A perfect square cannot equal a negative real number.

14. What is the missing term to complete the square: $x^2-\frac{7}{3}x+\square=\left(x-\frac{7}{6}\right)^2$?

    $\frac{49}{36}$. The missing term is $\left(\frac{-7/3}{2}\right)^2=\left(\frac{-7}{6}\right)^2=\frac{49}{36}$.

15. What equation results just before taking square roots when deriving the quadratic formula?

    $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$. This form leads directly to the quadratic formula when square roots are taken.

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

    $x=7$ or $x=-1$. Solving $x-3=4$ and $x-3=-4$ gives these solutions.

17. What is the completed-square form of $4x^2+4x-3=0$?

    $\left(x+\frac{1}{2}\right)^2=1$. First divide by 4, then complete the square on $x^2+x-\frac{3}{4}=0$.

18. What perfect-square identity rewrites $x^2+bx+\left(\frac{b}{2}\right)^2$?

    $\left(x+\frac{b}{2}\right)^2$. This factors as a perfect square with half the linear coefficient.

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

    $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$ squared makes the expression a perfect square trinomial.

20. What is the completed-square form of $x^2-8x+7=0$?

    $\left(x-4\right)^2=9$. Adding $\left(\frac{-8}{2}\right)^2=16$ to both sides, then simplifying.

21. What expression equals $\left(x+p\right)^2$ when expanded?

    $x^2+2px+p^2$. This is the expanded form of the perfect square $(x+p)^2$.

22. What is the completed-square form of $x^2+6x+13=0$?

    $\left(x+3\right)^2=-4$. Completing the square: $(x+3)^2=9-13=-4$.

23. What expression equals $\left(x-p\right)^2$ when expanded?

    $x^2-2px+p^2$. This is the expanded form of the perfect square $(x-p)^2$.

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

    Divide by $a$ so the $x^2$ coefficient is $1$. This makes the $x^2$ coefficient 1, simplifying the completing process.

25. What is the missing term to complete the square: $x^2+\frac{5}{2}x+\square=\left(x+\frac{5}{4}\right)^2$?

    $\frac{25}{16}$. The missing term is $\left(\frac{5/2}{2}\right)^2=\left(\frac{5}{4}\right)^2=\frac{25}{16}$.

26. What is the value of $p$ in $\left(x-p\right)^2=q$ in terms of the linear coefficient $b$ of $x^2+bx+c$?

    $p=-\frac{b}{2}$. The value of $p$ is negative half the linear coefficient.

27. What are the solutions of $x^2-8x+7=0$?

    $x=1$ or $x=7$. From $(x-4)^2=9$: $x-4=\pm 3$, so $x=4\pm 3$.

28. What is the missing term to complete the square: $x^2-12x+\square=\left(x-6\right)^2$?

    $36$. The missing term is $\left(\frac{-12}{2}\right)^2=(-6)^2=36$.

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

    $(x-p)^2=q$. This is the target form with a perfect square equal to a constant.

30. What is the completed-square form of $5x^2+20x+15=0$?

    $\left(x+2\right)^2=1$. First divide by 5: $x^2+4x+3=0$, then complete the square.

31. What is the completed-square form of $x^2-2x-8=0$?

    $\left(x-1\right)^2=9$. Completing the square: $(x-1)^2=1+8=9$.

32. What are the solutions of $2x^2+8x+6=0$?

    $x=-1$ or $x=-3$. From $(x+2)^2=1$: $x+2=\pm 1$, so $x=-2\pm 1$.

33. What is the completed-square form of $2x^2-4x-1=0$?

    $\left(x-1\right)^2=\frac{3}{2}$. First divide by 2: $x^2-2x-\frac{1}{2}=0$, then complete the square.

34. What is the completed-square form of $x^2+2x+10=0$?

    $\left(x+1\right)^2=-9$. Completing the square gives a perfect square equal to a negative number.

35. What does a negative discriminant $b^2-4ac<0$ indicate about the solutions?

    No real solutions (two complex solutions). Negative discriminant means taking the square root of a negative number.

36. What does a zero discriminant $b^2-4ac=0$ indicate about the solutions?

    One real solution (a double root). Zero discriminant means the square root is zero, giving one solution.

37. What property justifies dividing both sides by $a\neq 0$ in $ax^2+bx+c=0$?

    Division Property of Equality. This allows dividing both sides by the same nonzero value.

38. What is the completed-square form of $2x^2+8x+6=0$?

    $\left(x+2\right)^2=1$. First divide by 2, then complete the square on $x^2+4x+3=0$.

39. What is the vertex form of $y=x^2+bx+c$ after completing the square?

    $y=\left(x+\frac{b}{2}\right)^2+\left(c-\frac{b^2}{4}\right)$. This is vertex form with the vertex at $(-\frac{b}{2}, c-\frac{b^2}{4})$.

40. What is $p$ and $q$ if $x^2+10x+7=0$ is written as $\left(x-p\right)^2=q$?

    $p=-5, q=18$. From $x^2+10x+7=0$: $(x+5)^2=25-7=18$.

41. What is the key added term after dividing by $a$ in $ax^2+bx+c=0$ to complete the square?

    $\left(\frac{b}{2a}\right)^2$. After dividing by $a$, add half of the new linear coefficient squared.

42. What is the completed-square form of $x^2-10x+1=0$?

    $\left(x-5\right)^2=24$. Adding $\left(\frac{-10}{2}\right)^2=25$ to both sides, then simplifying.

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

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. This formula solves any quadratic equation and comes from completing the square.

44. What does a positive discriminant $b^2-4ac>0$ indicate about the solutions?

    Two distinct real solutions. A positive discriminant means the square root is real with two values.

45. What is the correct square-root step for $\left(x-3\right)^2=16$?

    $x-3=\pm 4$. Taking the square root of both sides: $\sqrt{16} = 4$.

46. What are the solutions of $3x^2-12x+9=0$?

    $x=1$ or $x=3$. From $(x-2)^2=1$: $x-2=\pm 1$, so $x=2\pm 1$.

47. What is the completed-square form of $x^2+6x+5=0$?

    $\left(x+3\right)^2=4$. Adding $\left(\frac{6}{2}\right)^2=9$ to both sides, then moving constants.

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

    $b^2-4ac$. This expression determines the nature of the quadratic's solutions.

49. What are the solutions of $4x^2+4x-3=0$?

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

50. What are the solutions of $x^2-10x+1=0$?

    $x=5\pm 2\sqrt{6}$. From $(x-5)^2=24$: $x=5\pm\sqrt{24}=5\pm 2\sqrt{6}$.

51. What expression results after completing the square on $x^2+\frac{b}{a}x$?

    $\left(x+\frac{b}{2a}\right)^2$. This is the perfect square formed after adding $\left(\frac{b}{2a}\right)^2$.

52. What are the solutions of $\left(x+2\right)^2=9$?

    $x=1$ or $x=-5$. From $(x+2)^2=9$: $x+2=\pm 3$, so $x=-2\pm 3$.

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