Projectile Motion Problem

A ball is thrown upward from ground level with an initial velocity of 48 feet per second. Its height $h$ (in feet) after $t$ seconds is given by $h = -16t^2 + 48t$. When will the ball hit the ground?

Understand what we need to find: The ball hits the ground when its height $h = 0$ = Set $h = 0$

Write the equation: $0 = -16t^2 + 48t$ = $-16t^2 + 48t = 0$

Factor out the GCF: $-16t(t - 3) = 0$ = Factored form

Apply zero product property: $-16t = 0$ or $t - 3 = 0$ = $t = 0$ or $t = 3$

Interpret the solutions: $t = 0$ is when the ball is thrown; $t = 3$ is when it lands = The ball lands at $t = 3$ seconds

Answer: The ball will hit the ground after 3 seconds.

Area Problem

A rectangular garden has a length that is 4 meters more than its width. If the area of the garden is 60 square meters, find the dimensions of the garden.

Define the variable: Let $w$ = width of the garden (in meters) = Width = $w$

Express length in terms of width: Length is 4 more than width = Length = $w + 4$

Write the area equation: Area = length $\times$ width, so $w(w + 4) = 60$ = $w^2 + 4w = 60$

Rewrite in standard form: $w^2 + 4w - 60 = 0$ = Standard form

Factor the quadratic: Find two numbers that multiply to $-60$ and add to $4$: $10$ and $-6$ = $(w + 10)(w - 6) = 0$

Solve for $w$: $w + 10 = 0$ or $w - 6 = 0$ = $w = -10$ or $w = 6$

Choose the valid solution: Width cannot be negative, so $w = 6$ = Width = 6 m, Length = 10 m

Answer: The garden is 6 meters wide and 10 meters long.

Number Problem

The product of two consecutive positive integers is 182. Find the integers.

Define the variable: Let $n$ = the first integer = First integer = $n$

Express the second integer: Consecutive means the next number = Second integer = $n + 1$

Write the equation: Product is 182, so $n(n + 1) = 182$ = $n^2 + n = 182$

Rewrite in standard form: $n^2 + n - 182 = 0$ = Standard form

Factor the quadratic: Find two numbers that multiply to $-182$ and add to $1$: $14$ and $-13$ = $(n + 14)(n - 13) = 0$

Solve for $n$: $n = -14$ or $n = 13$ = Two possible values

Choose the valid solution: We need positive integers, so $n = 13$ = The integers are 13 and 14

Answer: The two consecutive positive integers are 13 and 14.

Maximum Height Problem

A rocket is launched and its height (in meters) after $t$ seconds is given by $h = -5t^2 + 30t + 10$. What is the maximum height reached by the rocket?

Identify the type of problem: This is a parabola opening downward ($a = -5 < 0$), so it has a maximum = Find the vertex

Find the time at maximum height: Use $t = -\frac{b}{2a} = -\frac{30}{2(-5)} = -\frac{30}{-10}$ = $t = 3$ seconds

Calculate the maximum height: $h = -5(3)^2 + 30(3) + 10$ = $h = -5(9) + 90 + 10$

Simplify: $h = -45 + 90 + 10 = 55$ = $h = 55$ meters

Answer: The rocket reaches a maximum height of 55 meters (at $t = 3$ seconds).

Mistake: Forgetting to check if solutions make sense in context

Why: Mathematical solutions like negative lengths or times before launch are invalid in real-world problems.

Correct: Always interpret your answers: widths must be positive, time must be non-negative, and quantities must be reasonable.

Mistake: Setting up the equation incorrectly

Why: Misreading the relationship between quantities (e.g., confusing 'more than' with 'times').

Correct: Read carefully: '4 more than width' means $w + 4$, while '4 times the width' means $4w$.

Mistake: Using only one solution when both are valid

Why: Some problems have two meaningful solutions that both answer the question.

Correct: Check both solutions against the original problem. For example, a ball reaches 20 feet on the way up AND on the way down.

Mistake: Confusing the vertex formula with the quadratic formula

Why: The vertex formula $x = -\frac{b}{2a}$ finds the axis of symmetry, not the roots.

Correct: Use the vertex formula for maximum/minimum problems; use the quadratic formula to find where the parabola crosses the x-axis.

Sports and Projectile Motion

Athletes and coaches use quadratic equations to analyze the trajectory of balls, jumps, and throws.

A basketball player shoots from the three-point line. The ball's height follows $h = -0.05x^2 + 1.2x + 6$, where $x$ is the horizontal distance. Does the ball clear the 10-foot rim at $x = 20$ feet?

Architecture and Design

Architects use quadratic equations to design arches, domes, and optimize space.

A parabolic arch has the equation $y = -0.1x^2 + 2x$ where $x$ and $y$ are in meters. The arch spans from $x = 0$ to $x = 20$, and its maximum height is at the vertex: $y = 10$ meters.

Business and Economics

Companies use quadratic models to maximize profit and revenue.

A company finds that its profit (in thousands of dollars) from selling $x$ hundred items is $P = -2x^2 + 24x - 54$. Maximum profit occurs when $x = 6$ hundred items.

Q: How do I know when to use a quadratic equation?

A: Look for key indicators: area of rectangles (length times width), products of related quantities, projectile motion (objects thrown or dropped), or optimization (finding maximum or minimum values). If the problem involves multiplying a variable by itself, it's likely quadratic.

Q: What if I get two positive solutions?

A: Both might be valid! For example, a ball reaches a certain height twice: once going up and once coming down. Read the problem carefully to determine if you need one or both solutions.

Q: When should I use the quadratic formula vs. factoring?

A: Try factoring first if the numbers are simple. Use the quadratic formula when: the equation doesn't factor easily, you have decimals or fractions, or you want to be certain you find all solutions.