Factor Quadratics to Find Zeros - Algebra 2

1. What are the zeros of $f(x)=x^2-13x+36$?

   $x=4,9$. From factored form $x^2-13x+36=(x-4)(x-9)$.

2. What is the relationship between factors and zeros for $f(x)=(x-r)(x-s)$?

   Zeros are $x=r$ and $x=s$. When $(x-r)=0$ then $x=r$, and when $(x-s)=0$ then $x=s$.

3. What is the factored form of $x^2-6x+9$?

   $(x-3)^2$. Perfect square trinomial: $x^2-6x+9=(x-3)^2$.

4. What are the zeros of $f(x)=2x^2-5x-3$?

   $x=-\frac{1}{2},3$. From factored form $2x^2-5x-3=(2x+1)(x-3)$.

5. What are the zeros of $f(x)=(x-5)(x+2)$?

   $x=5,-2$. Set each factor equal to zero: $x-5=0$ or $x+2=0$.

6. What is the factored form of $x^2+7x+12$?

   $(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$: $3$ and $4$.

7. What are the zeros of $f(x)=3x^2-10x-8$?

   $x=-\frac{2}{3},4$. From factored form $3x^2-10x-8=(3x+2)(x-4)$.

8. What are the zeros of $f(x)=-(x-1)(x+6)$?

   $x=1,-6$. The negative sign doesn't affect the zeros, only the function's orientation.

9. What is the factored form of $2x^2+7x+3$?

   $(2x+1)(x+3)$. Use AC method: $2 \cdot 3 = 6$, factors $1$ and $6$ add to $7$.

10. What is the factored form of $x^2-16$?

    $(x-4)(x+4)$. Difference of squares: $x^2-16=(x-4)(x+4)$.

11. What are the zeros of $f(x)=(x+3)(x+8)$?

    $x=-3,-8$. Set each factor equal to zero: $x+3=0$ or $x+8=0$.

12. What is the factored form of $x^2-5x+6$?

    $(x-2)(x-3)$. Find two numbers that multiply to $6$ and add to $-5$: $-2$ and $-3$.

13. What is the factored form of $x^2-9x+20$?

    $(x-4)(x-5)$. Find two numbers that multiply to $20$ and add to $-9$: $-4$ and $-5$.

14. What is the factored form of $3x^2-12x$?

    $3x(x-4)$. Factor out the GCF of $3x$.

15. What is the factored form of $x^2+13x+36$?

    $(x+4)(x+9)$. Find two numbers that multiply to $36$ and add to $13$: $4$ and $9$.

16. What is the factored form of $x^2+11x+24$?

    $(x+3)(x+8)$. Find two numbers that multiply to $24$ and add to $11$: $3$ and $8$.

17. What are the zeros of $f(x)=(3x+2)(x+3)$?

    $x=-\frac{2}{3},-3$. Set each factor equal to zero: $3x+2=0$ or $x+3=0$.

18. What are the zeros of $f(x)=x^2-5x+6$?

    $x=2,3$. From factored form $x^2-5x+6=(x-2)(x-3)$.

19. What are the zeros of $f(x)=x^2-16$?

    $x=4,-4$. From factored form $x^2-16=(x-4)(x+4)$.

20. What is the factored form of $2x^2-5x-3$?

    $(2x+1)(x-3)$. Use AC method: $2 \cdot (-3) = -6$, factors $-6$ and $1$ add to $-5$.

21. What is the factored form of $x^2-8x+7$?

    $(x-1)(x-7)$. Find two numbers that multiply to $7$ and add to $-8$: $-1$ and $-7$.

22. What are the zeros of $f(x)=2x(x+5)$?

    $x=0,-5$. Set each factor equal to zero: $2x=0$ or $x+5=0$.

23. What are the zeros of $f(x)=3x(x-4)$?

    $x=0,4$. Set each factor equal to zero: $3x=0$ or $x-4=0$.

24. What are the zeros of $f(x)=x^2-8x+7$?

    $x=1,7$. From factored form $x^2-8x+7=(x-1)(x-7)$.

25. What is the factored form of $x^2-11x+24$?

    $(x-3)(x-8)$. Find two numbers that multiply to $24$ and add to $-11$: $-3$ and $-8$.

26. What is the factored form of $x^2-4x-21$?

    $(x-7)(x+3)$. Find two numbers that multiply to $-21$ and add to $-4$: $-7$ and $3$.

27. What are the zeros of $f(x)=x^2-9$?

    $x=3,-3$. Set each factor equal to zero: $x=3$ or $x=-3$.

28. What is the factored form of $x^2-13x+36$?

    $(x-4)(x-9)$. Find two numbers that multiply to $36$ and add to $-13$: $-4$ and $-9$.

29. What is the factored form of $3x^2+11x+6$?

    $(3x+2)(x+3)$. Use AC method: $3 \cdot 6 = 18$, factors $2$ and $9$ add to $11$.

30. What is the zero (with multiplicity) of $f(x)=(x-3)^2$?

    $x=3$ with multiplicity $2$. A repeated factor gives a zero with multiplicity equal to the exponent.

31. What are the zeros of $f(x)=(x+3)^2$?

    $x=-3$ (double root). From $(x+3)^2=0$, we get $x=-3$ with multiplicity $2$.

32. What are the zeros of $f(x)=(x-4)(x-5)$?

    $x=4,5$. Set each factor equal to zero: $x-4=0$ or $x-5=0$.

33. What is the factored form of $x^2-9$?

    $(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.

34. What is the factored form of $x^2-3x-10$?

    $(x-5)(x+2)$. Find two numbers that multiply to $-10$ and add to $-3$: $-5$ and $2$.

35. What is the factored form of $x^2+6x+9$?

    $(x+3)^2$. Perfect square trinomial: $x^2+6x+9=(x+3)^2$.

36. What are the zeros of $f(x)=(x+5)(x-3)$?

    $x=-5,3$. Set each factor equal to zero: $x+5=0$ or $x-3=0$.

37. What is the standard factored form of a quadratic with zeros $r$ and $s$ and leading coefficient $a$?

    $a(x-r)(x-s)$. General form where $a$ is the leading coefficient and $r,s$ are zeros.

38. What is the factored form of $x^2-2x-8$?

    $(x-4)(x+2)$. Find two numbers that multiply to $-8$ and add to $-2$: $-4$ and $2$.

39. What is the factored form of $x^2+2x-15$?

    $(x+5)(x-3)$. Find two numbers that multiply to $-15$ and add to $2$: $5$ and $-3$.

40. What are the zeros of $f(x)=(2x+1)(x+3)$?

    $x=-\frac{1}{2},-3$. Set each factor equal to zero: $2x+1=0$ or $x+3=0$.

41. What is the factored form of $x^2+4x-21$?

    $(x+7)(x-3)$. Find two numbers that multiply to $-21$ and add to $4$: $7$ and $-3$.

42. What are the zeros of $f(x)=x^2-11x+24$?

    $x=3,8$. From factored form $x^2-11x+24=(x-3)(x-8)$.

43. What are the zeros of $f(x)=(x-7)(x+3)$?

    $x=7,-3$. Set each factor equal to zero: $x-7=0$ or $x+3=0$.

44. What is the factored form of $4x^2-25$?

    $(2x-5)(2x+5)$. Difference of squares: $(2x)^2-5^2=(2x-5)(2x+5)$.

45. What is the factored form of $2x^2+10x$?

    $2x(x+5)$. Factor out the GCF of $2x$.

46. What is the factored form of $x^2+8x+7$?

    $(x+1)(x+7)$. Find two numbers that multiply to $7$ and add to $8$: $1$ and $7$.

47. What are the zeros of $f(x)=4x^2-25$?

    $x=\frac{5}{2},-\frac{5}{2}$. Set each factor equal to zero: $2x-5=0$ or $2x+5=0$.

48. What is the factored form of $3x^2-10x-8$?

    $(3x+2)(x-4)$. Use AC method: $3 \cdot (-8) = -24$, factors $2$ and $-12$ add to $-10$.

49. What is the factored form of $x^2+9x+20$?

    $(x+4)(x+5)$. Find two numbers that multiply to $20$ and add to $9$: $4$ and $5$.

50. What is the factored form of $x^2+x-12$?

    $(x+4)(x-3)$. Find two numbers that multiply to $-12$ and add to $1$: $4$ and $-3$.

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