Real-world examples of quadratic equations solved by various methods

📚 Quick Study Guide

• 📈 A quadratic equation is a polynomial equation of the second degree, generally represented as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
• 🔑 Factoring: Express the quadratic equation as a product of two binomials. If $ax^2 + bx + c = (px + q)(rx + s)$, then set each factor equal to zero and solve for $x$.
• ➕ Completing the Square: Transform the equation into the form $(x + h)^2 = k$, then solve for $x$ by taking the square root of both sides.
• ➗ Quadratic Formula: Use the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions of any quadratic equation.
• 🎯 The discriminant, $b^2 - 4ac$, determines the nature of the roots: positive (two real roots), zero (one real root), or negative (two complex roots).
• 💡 Applications: Quadratic equations are used to model projectile motion, areas, optimization problems, and many other real-world scenarios.

Practice Quiz

1. A ball is thrown vertically upward with an initial velocity of 48 ft/s. The height $h$ (in feet) of the ball after $t$ seconds is given by $h = 48t - 16t^2$. At what time $t$ will the ball hit the ground?
   1. $t = 1$ second
   2. $t = 2$ seconds
   3. $t = 3$ seconds
   4. $t = 4$ seconds
2. A rectangular garden has a length that is 5 feet longer than its width. If the area of the garden is 300 square feet, what is the width of the garden?
   1. 10 feet
   2. 12 feet
   3. 15 feet
   4. 20 feet
3. Solve the equation $x^2 - 5x + 6 = 0$ by factoring.
   1. $x = 1, 6$
   2. $x = -2, -3$
   3. $x = 2, 3$
   4. $x = -1, -6$
4. Solve the equation $2x^2 + 3x - 2 = 0$ using the quadratic formula.
   1. $x = 2, -1/2$
   2. $x = -2, 1/2$
   3. $x = 1, -2$
   4. $x = -1, 2$
5. A farmer wants to fence a rectangular area next to a river. He has 600 feet of fencing and does not need to fence along the river. What are the dimensions of the rectangle that maximize the enclosed area?
   1. 150 ft x 150 ft
   2. 100 ft x 400 ft
   3. 150 ft x 300 ft
   4. 100 ft x 200 ft
6. Solve the equation $x^2 + 4x - 5 = 0$ by completing the square.
   1. $x = 1, -5$
   2. $x = -1, 5$
   3. $x = -1, -5$
   4. $x = 1, 5$
7. The profit $P$ (in dollars) from selling $x$ units of a product is given by $P = -0.1x^2 + 50x - 1000$. How many units must be sold to maximize profit?
   1. 100 units
   2. 200 units
   3. 250 units
   4. 300 units