Test questions on applying quadratic factoring to real-life situations.

📚 Quick Study Guide

• 📐 Quadratic Expression: A quadratic expression is of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
• 🔍 Factoring: Factoring involves breaking down a quadratic expression into the product of two linear expressions, i.e., $ax^2 + bx + c = (px + q)(rx + s)$.
• 💡 Zero Product Property: If $(px + q)(rx + s) = 0$, then either $px + q = 0$ or $rx + s = 0$. This helps find the solutions (roots) of the quadratic equation.
• 📝 Real-Life Applications: Quadratic equations are used to model projectile motion, areas, and optimization problems. For example, the height of a ball thrown upwards can be modeled by a quadratic equation.
• 🧮 Finding Factors: Look for two numbers that multiply to $ac$ and add up to $b$ to factor $ax^2+bx+c$.
• 🌱 Perfect Square Trinomials: Recognize patterns like $a^2 + 2ab + b^2 = (a+b)^2$ or $a^2 - 2ab + b^2 = (a-b)^2$ to simplify factoring.

Practice Quiz

1. A rectangular garden has an area of $x^2 + 7x + 12$ square feet. If the length is $(x+4)$ feet, what is the width of the garden?
   1. (A) $x + 1$
   2. (B) $x + 2$
   3. (C) $x + 3$
   4. (D) $x + 5$
2. The height $h$ (in feet) of a projectile after $t$ seconds is given by $h = -16t^2 + 80t$. At what time $t$ will the projectile hit the ground (i.e., when $h=0$)?
   1. (A) 4 seconds
   2. (B) 5 seconds
   3. (C) 6 seconds
   4. (D) 8 seconds
3. A farmer wants to fence a rectangular area. He has 80 meters of fencing material. If one side is $x$ meters, the area is given by $A = x(40-x)$. Which of the following represents the factored form of $A$?
   1. (A) $x(x-40)$
   2. (B) $-x(x-40)$
   3. (C) $x(40+x)$
   4. (D) $-x(x+40)$
4. The product of two consecutive integers is 72. Which quadratic equation can be used to find the integers?
   1. (A) $x^2 + x - 72 = 0$
   2. (B) $x^2 - x - 72 = 0$
   3. (C) $x^2 + 72x = 0$
   4. (D) $x^2 - 72 = 0$
5. A small business's profit $P$ can be modeled by $P = -x^2 + 10x - 9$, where $x$ is the number of items sold. At what values of $x$ does the business break even (i.e., $P=0$)?
   1. (A) $x = 1$ and $x = 9$
   2. (B) $x = -1$ and $x = -9$
   3. (C) $x = 0$ and $x = 10$
   4. (D) $x = 2$ and $x = 8$
6. The area of a square is increased by adding 3 inches to one side and 8 inches to the other, resulting in a rectangle with an area of $x^2 + 11x + 24$. What was the original side length of the square?
   1. (A) 3 inches
   2. (B) 5 inches
   3. (C) 8 inches
   4. (D) $x$ inches
7. The number of diagonals, $d$, in a polygon with $n$ sides is given by $d = \frac{1}{2}n^2 - \frac{3}{2}n$. If a polygon has 20 diagonals, how many sides does it have?
   1. (A) 5 sides
   2. (B) 8 sides
   3. (C) 10 sides
   4. (D) 20 sides