Question

name: camila barbaperiod:algebraunit 8 hw 4 - finding the zeros with factoringdirections: show all work in order to receive full credit.1. find the zeros of $f(x) = (2x - 1)(x - 5)$2. find the zeros of $f(x) = x^2 - 11x + 24$3. find the zeros of $g(x) = x^2 + 2x - 24$4. find the zeros of $g(x) = 25x^2 - 81$5. find the zeros of $h(x) = 3x^2 - 27$6. find the zeros of $h(x) = 2x^2 + 8x + 6$

Explanation:

1. Problem 1: Set factors to 0

Set each linear factor equal to 0.
$2x - 1 = 0$ and $x - 5 = 0$

1. Problem 1: Solve for x

Solve each equation for x.
$2x=1 \implies x=\frac{1}{2}$; $x=5$

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2. Problem 2: Factor quadratic

Factor $x^2-11x+24$ into binomials.
$x^2-11x+24=(x-3)(x-8)$

2. Problem 2: Set factors to 0

Set each factor equal to 0 and solve.
$x-3=0 \implies x=3$; $x-8=0 \implies x=8$

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3. Problem 3: Factor quadratic

Factor $x^2+2x-24$ into binomials.
$x^2+2x-24=(x+6)(x-4)$

3. Problem 3: Set factors to 0

Set each factor equal to 0 and solve.
$x+6=0 \implies x=-6$; $x-4=0 \implies x=4$

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4. Problem 4: Factor difference of squares

Factor $25x^2-81$ (difference of squares: $a^2-b^2=(a-b)(a+b)$).
$25x^2-81=(5x-9)(5x+9)$

4. Problem 4: Set factors to 0

Set each factor equal to 0 and solve.
$5x-9=0 \implies x=\frac{9}{5}$; $5x+9=0 \implies x=-\frac{9}{5}$

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5. Problem 5: Factor out GCF first

Factor out 3, then factor difference of squares.
$3x^2-27=3(x^2-9)=3(x-3)(x+3)$

5. Problem 5: Set factors to 0

Set non-constant factors equal to 0 and solve.
$x-3=0 \implies x=3$; $x+3=0 \implies x=-3$

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6. Problem 6: Factor out GCF first

Factor out 2, then factor the quadratic.
$2x^2+8x+6=2(x^2+4x+3)=2(x+1)(x+3)$

6. Problem 6: Set factors to 0

Set non-constant factors equal to 0 and solve.
$x+1=0 \implies x=-1$; $x+3=0 \implies x=-3$

Answer:

1. $x=\frac{1}{2}$ and $x=5$
2. $x=3$ and $x=8$
3. $x=-6$ and $x=4$
4. $x=\frac{9}{5}$ and $x=-\frac{9}{5}$
5. $x=3$ and $x=-3$
6. $x=-1$ and $x=-3$