9.4 Writing and Modeling Real-World Problems with Quadratic Equations

Why Writing and Modeling Quadratic Context Problems Matters

Indicator A.4.b ("Write one-variable quadratic equations to represent context") is the applied half of quadratic equations on the GED. Rather than handing you an equation to solve, these items describe a real situation — a ball thrown in the air, a company's profit over time, or a rectangular area — and ask you to build the correct quadratic equation, then interpret the solution in terms of the scenario. This is where quadratics connect to two of the GED's favorite real-world models: projectile motion and area/profit modeling.

Model 1: Projectile Motion

The most frequently tested quadratic context on the GED is an object thrown, dropped, or launched into the air. The standard model is:

h(t) = −16t² + v₀t + h₀

where:

• h(t) = height of the object (in feet) at time t (in seconds)
• −16 is a constant related to gravity's effect in feet per second squared (this coefficient is always −16 on the GED, since it uses feet and seconds)
• v₀ = initial velocity (how fast the object is launched upward)
• h₀ = initial height (how high off the ground the object starts)

Worked Example: A ball is thrown upward from ground level with the height modeled by h(t) = −16t² + 64t. At what time does the ball hit the ground?

• The ball "hits the ground" when height = 0, so set h(t) = 0: −16t² + 64t = 0
• Factor out the greatest common factor, −16t: −16t(t − 4) = 0
• Zero Product Property: t = 0 or t = 4
• Interpretation: t = 0 is the moment the ball is thrown (height is 0 at the start too), which is not the answer being asked for. The ball hits the ground at t = 4 seconds.

This example illustrates a key GED skill beyond the algebra: rejecting the solution that doesn't fit the real-world context. Quadratic equations mathematically produce two roots, but a word problem often only has one root that makes physical sense.

Model 2: Profit, Revenue, and Business Context

Quadratic models also appear in business scenarios where profit rises, peaks, and falls over time — the parabola's vertex represents the maximum (or minimum) value.

Worked Example: A company's profit (in thousands of dollars) is modeled by P(t) = −t² + 6t − 5, where t is the number of years since 2020. In which year did the company reach maximum profit?

• Since the leading coefficient (−1) is negative, the parabola opens downward, so its vertex is a maximum point.
• The t-coordinate of the vertex of ax² + bx + c is found with t = −b / (2a).
• Here a = −1, b = 6: t = −6 / (2 × −1) = −6 / −2 = 3
• Since t = 3 means 3 years after 2020, maximum profit occurred in 2023.

Model 3: Area Problems

A rectangle's area (length × width) creates a quadratic relationship whenever length and width are both expressed in terms of the same variable.

Worked Example: A rectangular garden's length is 3 feet more than its width, and its area is 54 square feet. Write an equation and find the width.

• Let w = width, so length = w + 3.
• Area equation: w(w + 3) = 54, which expands to w² + 3w − 54 = 0
• Factor: two numbers that multiply to −54 and add to 3 are 9 and −6: (w + 9)(w − 6) = 0
• Solutions: w = −9 or w = 6
• Interpretation: A width cannot be negative, so reject w = −9. The width is 6 feet (and the length is 9 feet).

Recognizing Which Model Applies

Common Traps to Avoid

• Reporting both mathematical roots without checking which one fits the real-world situation (negative time, negative length, or negative width are never valid answers).
• Confusing the vertex formula −b/(2a) with a root of the equation — the vertex tells you when the maximum/minimum occurs, not when the value is zero.
• Mixing up which variable is time vs. height (or width vs. length) when translating the word problem into the equation.
• Forgetting to add the starting year, starting height, or other offset back into the final answer after solving for t (e.g., t = 3 years after 2020 is the year 2023, not "year 3").

Key Takeaways

Writing quadratic equations from context is really two skills combined: correctly building the equation from the scenario (projectile, profit, or area), and correctly interpreting which of the two mathematical solutions actually answers the question — since negative time, negative length, and other impossible values must always be rejected. Practice identifying the scenario type from its keywords first, then apply the matching formula (h(t) = −16t² + v₀t + h₀ for projectiles, −b/2a for a vertex/maximum, or dimension × dimension for area) before solving.