Factor Quadratics to Find Zeros - Algebra 1

1. What are the zeros of $f(x)$ if $f(x)=(x-1)(x-9)$?

   $x=1$ and $x=9$. Set each factor equal to zero: $(x-1)=0$ and $(x-9)=0$.

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

   $x=4$ and $x=-4$. From $(x-4)(x+4)=0$, set each factor to zero.

3. What is the first step to factor $6x^2+9x$ to reveal zeros?

   Factor out the GCF: $3x(2x+3)$. Factor out the greatest common factor before finding zeros.

4. What is the factored form of $x^2-5x-14$?

   $(x-7)(x+2)$. Find two numbers that multiply to $-14$ and add to $-5$: $-7$ and 2.

5. What is the factored form of $4x^2-12x+9$?

   $(2x-3)^2$. Perfect square trinomial: $(2x-3)^2$.

6. What are the zeros of $f(x)=x^2-12x+35$ after factoring?

   $x=5$ and $x=7$. From $(x-5)(x-7)=0$, set each factor to zero.

7. 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 $-3$.

8. What are the zeros of $f(x)=x^2+12x+35$ after factoring?

   $x=-5$ and $x=-7$. From $(x+5)(x+7)=0$, set each factor to zero.

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

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

10. What are the zeros of $f(x)=x^2+7x+12$ after factoring?

    $x=-3$ and $x=-4$. From $(x+3)(x+4)=0$, set each factor to zero.

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

    $x=-\frac{3}{2}$ (double root). Square factor gives one zero with multiplicity 2.

12. What is the factored form of $x^2+5x-14$?

    $(x+7)(x-2)$. Find two numbers that multiply to $-14$ and add to 5: 7 and $-2$.

13. What is the factored form of $x^2-6x$?

    $x(x-6)$. Factor out GCF of $x$ from both terms.

14. What is the factored form of $x^2-12x+35$?

    $(x-5)(x-7)$. Find two numbers that multiply to 35 and add to $-12$: $-5$ and $-7$.

15. What are the zeros of $f(x)=x^2+3x-10$ after factoring?

    $x=-5$ and $x=2$. From $(x+5)(x-2)=0$, set each factor to zero.

16. What are the zeros of $f(x)$ if $f(x)=(3x+6)(x-5)$?

    $x=-2$ and $x=5$. Set each factor equal to zero: $(3x+6)=0$ gives $x=-2$.

17. 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.

18. What are the zeros of $f(x)=2x^2+8x$ using its factored form?

    $x=0$ and $x=-4$. From $2x(x+4)=0$, set each factor to zero.

19. What are the zeros of $f(x)$ if $f(x)=(5x-10)(2x+1)$?

    $x=2$ and $x=-\frac{1}{2}$. Set each factor equal to zero: $(5x-10)=0$ gives $x=2$.

20. What are the zeros of $f(x)=3x^2-12$ using the factored form?

    $x=2$ and $x=-2$. From $3(x-2)(x+2)=0$, only the linear factors give zeros.

21. What are the zeros of $f(x)=x^2-6x$ after factoring?

    $x=0$ and $x=6$. From $x(x-6)=0$, set each factor to zero.

22. What is the factored form of $x^2+12x+35$?

    $(x+5)(x+7)$. Find two numbers that multiply to 35 and add to 12: 5 and 7.

23. What are the zeros of $f(x)=3x^2+2x-8$ after factoring?

    $x=\frac{4}{3}$ and $x=-2$. From $(3x-4)(x+2)=0$, set each factor to zero.

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

    $2x(x+4)$. Factor out GCF of $2x$ from both terms.

25. What are the zeros of $f(x)=2x^2+5x+3$ after factoring?

    $x=-\frac{3}{2}$ and $x=-1$. From $(2x+3)(x+1)=0$, set each factor to zero.

26. 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.

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

    $x=\frac{3}{4}$ and $x=2$. Set each factor equal to zero: $(4x-3)=0$ gives $x=\frac{3}{4}$.

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

    $3(x-2)(x+2)$. Factor out 3, then use difference of squares.

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

    $x=4$ (double root). Square factor gives one zero with multiplicity 2.

30. 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.

31. What is the factored form of $x^2+10x+25$?

    $(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.

32. What are the zeros of $f(x)=2x^2-x-3$ after factoring?

    $x=\frac{3}{2}$ and $x=-1$. From $(2x-3)(x+1)=0$, set each factor to zero.

33. What is the factored form of $2x^2+5x+3$?

    $(2x+3)(x+1)$. Use AC method or trial factoring with leading coefficient 2.

34. What are the zeros of $f(x)=x^2-9x+20$ after factoring?

    $x=5$ and $x=4$. From $(x-5)(x-4)=0$, set each factor to zero.

35. What property connects factors to zeros in $f(x)=a(x-r_1)(x-r_2)$?

    Zero product property: $f(x)=0\Rightarrow x=r_1$ or $x=r_2$. If a product equals zero, at least one factor must be zero.

36. 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 $-2$.

37. What are the zeros of $f(x)=x^2+5x-14$ after factoring?

    $x=-7$ and $x=2$. From $(x+7)(x-2)=0$, set each factor to zero.

38. What are the zeros of $f(x)=x^2+10x+25$ after factoring?

    $x=-5$ (double root). From $(x+5)^2=0$, the repeated factor gives one zero.

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

    $(2x-3)(x+1)$. Use AC method or trial factoring with leading coefficient 2.

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

    $(x+6)(x-4)$. Find two numbers that multiply to $-24$ and add to 2: 6 and $-4$.

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

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

42. What is the factored form of $3x^2+2x-8$?

    $(3x-4)(x+2)$. Use AC method or trial factoring with leading coefficient 3.

43. What are the zeros of $f(x)=x^2-5x-14$ after factoring?

    $x=7$ and $x=-2$. From $(x-7)(x+2)=0$, set each factor to zero.

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

    $(x-4)(x+4)$. Difference of squares pattern: $a^2-b^2=(a-b)(a+b)$.

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

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

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

    $x=-8$ and $x=-1$. Set each factor equal to zero: $(x+8)=0$ and $(x+1)=0$.

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

    $x=0$ and $x=-\frac{3}{2}$. Set each factor to zero: $3x=0$ and $(2x+3)=0$.

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