Multiple Choice

1. To solve the equation $x^2 = 81$ using the square root method, which of the following represents the correct solutions?

   $x= \pm 40.5$

   $x=9$

   $x= -9$

   $x = \pm 9$

2. What is the first step required to solve the equation $4x^2 = 100$ using the square root method?

   Isolate $x^2$ by dividing both sides by 4.

   Take the square root of both sides immediately.

   Subtract 100 from both sides.

   Take the square root of 4.

3. Solve the equation $2x^2 - 72 = 0$.

   $x= \pm 18$

   $x = \pm 6$

   $x= \pm 6$

   $x=36$

4. If $9x^2 = 36$, what are the solutions for $x$?

   $x= \pm 3$

   $x=4$

   $x= \pm 4$

   $x = \pm 2$

5. The equation $x^2 = 144$ has two solutions because when applying the square root method, one must consider both the positive and negative roots. What are the solutions?

   $x = 12$ and $x = 72$

   $x = 12$ and $x = -12$

   $x = 12$

   $x=144$ and $x=-144$

6. To solve $5x^2 = 245$, you first divide by 5, resulting in $x^2 = 49$. What is the final step?

   $x = 7$

   Take the square root of 49 and include the $\pm$ sign, resulting in $x = \pm 7$.

   Add 5 to both sides.

   Divide 49 by 2.

7. Find the solutions for the equation $\frac{1}{2}x^2 = 32$.

   $x = \pm 32$

   $x = 16$

   $x = \pm 4$

   $x = \pm 8$

8. Which equation yields the solutions $x = \pm 5$?

   $5x^2 = 10$

   $2x^2 = 18$

   $x^2 = 10$

   $x^2 - 25 = 0$

9. Consider the equation $3x^2 + 18 = 93$. What value must $x^2$ equal before applying the square root?

   $x^2 = 37$

   $x^2 = 25$

   $x^2 = 75$

10. Solve the equation $4x^2 - 1 = 0$.

    $x = \pm 0.5$

    $x = \pm 4$

    $x = \pm 1$

    $x = \pm \frac{1}{2}$

11. When solving $x^2 = 40$, the constant 40 is not a perfect square. The answer must be simplified. Which option correctly simplifies the radical $\sqrt{40}$?

    $4\sqrt{10}$

    $10\sqrt{4}$

    $2\sqrt{10}$

    $\pm 2\sqrt{10}$

12. Solve the equation $x^2 = 72$ and express the solutions in simplest radical form.

    $x = \pm 6\sqrt{2}$

    $x = \pm 72$

    $x = \pm 2\sqrt{18}$

    $x = \pm 3\sqrt{8}$

13. Find the solutions to $3x^2 = 30$.

    $x = \pm \sqrt{30}$

    $x = \pm 3\sqrt{10}$

    $x = \pm \sqrt{10}$

    $x = \pm 10$

14. Solve $2x^2 - 50 = 20$. Express the answer in simplest radical form.

    $x = \pm \sqrt{35}$

    $x = \pm 5\sqrt{2}$

    $x = \pm 7.5$

15. What are the exact solutions to $x^2 - 48 = 0$?

    $x = \pm 2\sqrt{12}$

    $x = \pm 4\sqrt{3}$

    $x = \pm 4$

    $x = \pm 16\sqrt{3}$

16. Solve the equation $(x+5)^2 = 1$.

    $x=4$ and $x=6$

    $x = -6$

    $x = -4$

    $x = -4$ and $x = -6$

17. Apply the square root method to solve $(x-1)^2 = 36$.

    $x = 1$ and $x = -35$

    $x = 7$

    $x = 7$ and $x = -5$

18. Find the solutions for $(x+2)^2 = 8$. Express your answer using simplified radicals.

    $x = -2 \pm 4\sqrt{2}$

    $x = 2 \pm 2\sqrt{2}$

    $x = -2 \pm 2\sqrt{2}$

    $x = -2 \pm \sqrt{8}$

19. To solve $(x-4)^2 = 50$, you first take the square root of both sides. This gives $x-4 = \pm \sqrt{50}$. What is the final simplified solution?

    $x = -4 \pm 5\sqrt{2}$

    $x = 4 \pm 5\sqrt{2}$

    $x = 4 \pm 50$

    $x = 4 \pm 2\sqrt{25}$

20. What is the resulting equation after correctly isolating $x$ in $(x-10)^2 = 16$?

    $x = 10 \pm 16$

    $x = 14$

    $x = 14$ and $x = 6$

21. When factoring the trinomial $x^2 + 8x + 15$, we look for two numbers that multiply to 15 (the product) and add up to 8 (the sum). What are those two numbers?

    3 and 5

    10 and 5

    8 and 7

    1 and 15

22. Factor the trinomial $x^2 - 2x - 8$.

    $(x-4)(x-2)$

    $(x+8)(x-1)$

    $(x+4)(x-2)$

    $(x-4)(x+2)$

23. Factor the expression $x^2 + 10x + 9$.

    $(x-9)(x-1)$

    $(x-3)^2$

    $(x+9)(x+1)$

    $(x+10)(x+9)$

24. The product-sum method is used to factor $x^2 - 12x + 35$. The two factors used in the binomials are:

    -5 and -7

    5 and 7

    12 and -1

    -1 and -35

25. Identify the factored form of $x^2 + 4x - 12$.

    $(x-4)(x+3)$

    $(x-6)(x+2)$

    $(x-2)(x-6)$

    $(x+6)(x-2)$

26. Solve the quadratic equation $x^2 + 5x + 6 = 0$ by factoring.

    $x=-6$ and $x=1$

    $x=2$ and $x=3$

    $x = -2$ and $x = -3$

    $x=5$ and $x=1$

27. The Null Factor Law (or Zero Product Property) states that if $AB=0$, then $A=0$ or $B=0$. If you factor $x^2 - 4x - 5 = 0$ into $(x-5)(x+1) = 0$, what are the solutions?

    $x=5$ and $x=-1$

    $x=-5$ and $x=1$

    $x=-4$ and $x=-1$

    $x=0$ and $x=4$

28. Find the roots of the equation $x^2 - 10x + 16 = 0$.

    $x = 8$ and $x = 2$

    $x = 10$ and $x = 6$

    $x = -8$ and $x = -2$

    $x = 16$ and $x = 1$

29. Before applying the Null Factor Law, the equation must be set to zero. Find the solutions for $x^2 = 3x + 4$.

    $x=3$ and $x=4$

    $x=4$ and $x=-3$

    $x=4$ and $x=-1$

    $x=1$ and $x=4$

30. Which of the following equations has the solutions $x=7$ and $x=-3$?

    $x^2 - 21 = 0$

    $x^2 + 4x + 21 = 0$

    $x^2 - 4x - 21 = 0$

31. To factor the complex trinomial $2x^2 + 7x + 3$ using decomposition, we first look for two numbers that multiply to $a \cdot c = 6$ and add up to $b = 7$. What are these two numbers?

    3 and 4

    2 and 5

    1 and 6

32. Which expression correctly shows the decomposition step when factoring $3x^2 - 4x - 4$? (Hint: Product = -12, Sum = -4)

    $3x^2 - 6x + 2x - 4$

    $3x^2 - 3x - x - 4$

    $3x^2 + 12x - x - 4$

    $3x^2 - 6x - 2x - 4$

33. Factor the complex trinomial $2x^2 + 11x + 5$.

    $(2x+11)(x+5)$

    $(2x+5)(x+1)$

    $(x+5)(x+2)$

    $(2x+1)(x+5)$

34. Which binomial is a factor of $4x^2 + 8x + 3$?

    $(2x+1)$

    $(4x+1)$

    $(4x+3)$

35. Factor the expression $5x^2 - 13x + 6$.

    $(x-6)(5x-1)$

    $(5x+2)(x-3)$

    $(5x-3)(x-2)$

36. Solve the quadratic equation $3x^2 + 10x - 8 = 0$. (Hint: $3x^2 + 12x - 2x - 8 = 0$)

    $x = \frac{4}{3}$ and $x = 2$

    $x=4$ and $x= -2$

    $x = -4$ and $x = \frac{2}{3}$

    $x = -4$ and $x = 2$

37. If the factored form of a quadratic equation is $(2x+5)(x-1) = 0$, what are the specific solutions (roots) for $x$?

    $x = -\frac{5}{2}$ and $x = 1$

    $x = 2$ and $x = 5$

    $x = -5$ and $x = 1$

    $x = 5$ and $x = -1$

38. Find the solutions for the equation $4x^2 - 9x + 2 = 0$.

    $x=-4$ and $x=-2$

    $x = \frac{1}{4}$ and $x = 2$

    $x = -\frac{1}{4}$ and $x = 2$

    $x=4$ and $x=\frac{1}{2}$

39. Solve $2x^2 + 3x = 2$.

    $x=1$ and $x=-\frac{1}{2}$

    $x=-2$ and $x = \frac{1}{2}$

    $x=-2$ and $x=1$

    $x=2$ and $x=\frac{1}{2}$

40. The Null Factor Law requires that a factored expression equals zero. If $5x(x+1)(3x-2) = 0$, how many distinct solutions are there?

    4

    2

    1

    3

41. A Grade 9 student is solving the equation $3x^2 + 15 = 129$ using the square root method. After isolating the squared term, which of the following represents the completely simplified solution for $x$?

    $x = \pm 4\sqrt{2}$

    $x = \pm 6$

    $x = \pm \sqrt{38}$

    $x = \pm 2\sqrt{11}$

42. Determine the solutions for the equation $5(x-2)^2 - 8 = 32$. This problem requires isolating the binomial square before applying the square root property.

    $x = 0$ and $x = 4$

    $x = 2 \pm 2\sqrt{2}$

    $x = 6$ and $x = -2$

    $x = 4$ and $x = -2$

43. Solve the quadratic equation $(3x+2)^2 = 48$. Express the final solution using the simplified radical form, demonstrating competence in extending the square root method to binomial squares.

    $x = \frac{3 \pm 4\sqrt{3}}{2}$

    $x = -2 \pm 4\sqrt{3}$

    $x = \frac{-2 \pm 2\sqrt{3}}{3}$

    $x = \frac{-2 \pm 4\sqrt{3}}{3}$

44. Before applying the Null Factor Law, the equation $2x^2 - 22x + 60 = 0$ must first be factored completely. Identify the correct factored form, utilizing the Product-Sum method after factoring out the Greatest Common Factor (GCF).

    $2(x-5)(x-6)$

    $(2x-10)(x-6)$

    $2(x-5)(x+6)$

    $2(x-15)(x-2)$

45. Solve the equation $x^2 + 5x = 50$ using factoring and the Null Factor Law. What are the two roots of this quadratic equation?

    $x = 5$ and $x = 10$

    $x = -5$ and $x = -10$

    $x = -10$ and $x = 5$

    $x = 10$ and $x = -5$

46. The equation $3x^2 - 13x - 10 = 0$ must be solved by factoring using decomposition since the leading coefficient $a \neq 1$. Which pair of values represents the correct solutions (roots) for $x$?

    $x = 5/3$ and $x = -2$

    $x = 5$ and $x = -2/3$

    $x = 10/3$ and $x = -1$

47. A challenging application of the square root method involves the equation $-3(x^2 - 7) = 33$. Find the exact solutions for $x$.

    $x = \pm 2\sqrt{5}$

    $x = \pm \sqrt{18}$

    $x = \pm 2i$

    $x = \pm 4$

48. Factor the complex trinomial $12x^2 + 11x - 15$ completely using the method of decomposition.

    $(4x-3)(3x+5)$

    $(12x-5)(x+3)$

    $(6x-5)(2x+3)$

    $(4x+3)(3x-5)$

49. Find the solutions to the equation $9x^2 = 12x$ by first setting the equation to zero and then applying the Null Factor Law.

    $x = 3/4$ and $x = 0$

    $x = 4/3$ only

    $x = 4/3$ and $x = 0$

    $x = 12/9$ and $x = -12/9$

50. Factor the trinomial $-4x^2 - 4x + 15$ completely. This requires factoring out a negative GCF first and then using decomposition on the resulting trinomial.

    $-(2x-5)(2x+3)$

    $(-2x-3)(2x+5)$

    $-(2x+5)(2x-3)$

    $-(4x-5)(x+3)$