Question: User: Name: Gianna A Date: Period Review Worksheet (Wednesday 09/08/) Instructions: Solve each problem on a separate sheet of paper if additional space is needed. Show all work completely. Simplify all expressions and underline your final answers. Section 1: Factoring Quadratics (a = 1) 1. Factor completely: $x^2 + 9x + 20$ 2. Factor completely: $x^2 - 7x + 12$ 3. Factor completely: $x^2 + 2x - 15$ 4. Factor completely: $x^2 - 4x - 21$ 5. Factor completely: $x^2 - 16$ 6. Solve for $x$ by factoring: $x^2 - 11x + 24 = 0$ 7. Word Problem: The area of a rectangular garden is represented by the expression $x^2 + 8x + 12$ square feet. Write expressions in terms of $x$ for the possible length and width of the garden. Section 2: Linear Expressions & Equations 1. Simplify the linear expression: $4(2x - 5) - 3(x + 2)$ 2. Solve for $x$: $5x - 8 = 2x + 13$ 3. Solve for $x$: $3(x + 4) = 2(2x - 1)$ 4. Solve for $x$: $\frac{x}{4} + 5 = 11$ 5. Word Problem: A rideshare company charges a base fee of 3.50 plus 1.75 per mile driven. If Marcus spent a total of 24.50 on his trip, write and solve a linear equation to find the distance ($m$), in miles, of his trip.

1. **Problem:** Factor completely $x^2 + 9x + 20$. Formula: For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. Step 1: Find factors of 20 that add to 9: 4 and 5. Step 2: Write as $(x + 4)(x + 5)$. \underline{Answer:} $(x + 4)(x + 5)$. 2. **Problem:** Factor completely $x^2 - 7x + 12$. Step 1: Factors of 12 that add to -7: -3 and -4. Step 2: Write as $(x - 3)(x - 4)$. \underline{Answer:} $(x - 3)(x - 4)$. 3. **Problem:** Factor completely $x^2 + 2x - 15$. Step 1: Factors of -15 that add to 2: 5 and -3. Step 2: Write as $(x + 5)(x - 3)$. \underline{Answer:} $(x + 5)(x - 3)$. 4. **Problem:** Factor completely $x^2 - 4x - 21$. Step 1: Factors of -21 that add to -4: -7 and 3. Step 2: Write as $(x - 7)(x + 3)$. \underline{Answer:} $(x - 7)(x + 3)$. 5. **Problem:** Factor completely $x^2 - 16$. Step 1: Recognize difference of squares: $a^2 - b^2 = (a - b)(a + b)$. Step 2: $x^2 - 16 = (x - 4)(x + 4)$. \underline{Answer:} $(x - 4)(x + 4)$. 6. **Problem:** Solve for $x$ by factoring $x^2 - 11x + 24 = 0$. Step 1: Factor quadratic: factors of 24 that add to -11 are -8 and -3. Step 2: Write as $(x - 8)(x - 3) = 0$. Step 3: Set each factor to zero: $$x - 8 = 0 \Rightarrow x = 8$$ $$x - 3 = 0 \Rightarrow x = 3$$ \underline{Answer:} $x = 8$ or $x = 3$. 7. **Word Problem:** Area of garden is $x^2 + 8x + 12$. Step 1: Factor $x^2 + 8x + 12$. Step 2: Factors of 12 that add to 8: 6 and 2. Step 3: Length and width expressions: $(x + 6)$ and $(x + 2)$. \underline{Answer:} Length = $x + 6$, Width = $x + 2$. --- Section 2: 1. **Problem:** Simplify $4(2x - 5) - 3(x + 2)$. Step 1: Distribute: $$4 \times 2x = 8x$$ $$4 \times (-5) = -20$$ $$-3 \times x = -3x$$ $$-3 \times 2 = -6$$ Step 2: Combine: $$8x - 20 - 3x - 6 = (8x - 3x) + (-20 - 6) = 5x - 26$$ \underline{Answer:} $5x - 26$. 2. **Problem:** Solve $5x - 8 = 2x + 13$. Step 1: Subtract $2x$ from both sides: $$5x - \cancel{2x} - 8 = \cancel{2x} + 13 - 2x \Rightarrow 3x - 8 = 13$$ Step 2: Add 8 to both sides: $$3x - 8 + 8 = 13 + 8 \Rightarrow 3x = 21$$ Step 3: Divide both sides by 3: $$\frac{3x}{\cancel{3}} = \frac{21}{\cancel{3}} \Rightarrow x = 7$$ \underline{Answer:} $x = 7$. 3. **Problem:** Solve $3(x + 4) = 2(2x - 1)$. Step 1: Distribute: $$3x + 12 = 4x - 2$$ Step 2: Subtract $3x$ from both sides: $$3x - \cancel{3x} + 12 = 4x - \cancel{3x} - 2 \Rightarrow 12 = x - 2$$ Step 3: Add 2 to both sides: $$12 + 2 = x - 2 + 2 \Rightarrow 14 = x$$ \underline{Answer:} $x = 14$. 4. **Problem:** Solve $\frac{x}{4} + 5 = 11$. Step 1: Subtract 5 from both sides: $$\frac{x}{4} + 5 - 5 = 11 - 5 \Rightarrow \frac{x}{4} = 6$$ Step 2: Multiply both sides by 4: $$4 \times \frac{x}{4} = 6 \times 4 \Rightarrow x = 24$$ \underline{Answer:} $x = 24$. 5. **Word Problem:** Base fee 3.50 plus 1.75 per mile, total 24.50. Step 1: Write equation: $$3.50 + 1.75m = 24.50$$ Step 2: Subtract 3.50 from both sides: $$1.75m = 24.50 - 3.50 = 21$$ Step 3: Divide both sides by 1.75: $$m = \frac{21}{1.75}$$ Step 4: Simplify fraction: $$m = 12$$ \underline{Answer:} The trip distance is 12 miles.