Factor This Equation -16t2 64t 80

You're staring at a quadratic. Consider this: maybe it showed up on a homework assignment. Maybe it's modeling the height of a ball thrown off a building. Either way, the expression -16t² + 64t + 80 is sitting there, unfactored, and you need to break it down.

Let's do that right now. Then we'll talk about why it matters, where people trip up, and what this particular quadratic is actually telling you about the physical world.

What Is This Expression

First, let's be clear about what we're looking at.

-16t² + 64t + 80 is a quadratic expression in standard form: at² + bt + c. The variable is t, which almost always stands for time in seconds when you see this specific coefficient pattern. The -16 is the giveaway — that's half of -32, the acceleration due to gravity in feet per second squared.

So this isn't just an abstract algebra exercise. Which means it's a height function. Specifically, it models the height (in feet) of an object launched upward at 64 ft/s from an initial height of 80 feet.

The three terms each have a job:

• -16t² — gravity pulling the object down
• +64t — the initial upward velocity pushing it up
• +80 — the starting height

When you factor this, you're finding the t-values where height equals zero. In other words: when does the object hit the ground?

The Factoring Process Step by Step

Here's the clean version. Then I'll show the version with all the scratch work.

Final answer: -16(t - 5)(t + 1)

Step 1: Factor out the GCF

Every coefficient is divisible by 16. The leading coefficient is negative, so factor out -16 — this makes the trinomial inside easier to work with.

-16t² + 64t + 80 = -16(t² - 4t - 5)

Notice the sign flips inside. -16 times -4t gives +64t. Also, -16 times -5 gives +80. Check passed.

Step 2: Factor the trinomial

Now you need two numbers that multiply to -5 and add to -4.

That's -5 and +1.

-5 × 1 = -5 -5 + 1 = -4

So t² - 4t - 5 = (t - 5)(t + 1)

Step 3: Put it back together

-16(t - 5)(t + 1)

Done.

Why the Factored Form Matters

The standard form tells you the initial conditions. The factored form tells you the events*.

Set -16(t - 5)(t + 1) = 0 and you get two solutions: t = 5 and t = -1.

t = 5 seconds — the object hits the ground. This is the physical answer.

t = -1 seconds — mathematically valid, physically meaningless. It represents the time before* launch when the object would have been at ground level if its trajectory had been extended backward. You discard it in context, but it's still a real root of the polynomial.

Factoring is worth taking seriously — and now you know why. It converts a continuous curve into discrete, meaningful moments.

What This Quadratic Actually Represents

Let's visualize the full picture.

At t = 0: height = 80 feet. The object starts on a rooftop or cliff edge.

At t = 1: height = -16 + 64 + 80 = 128 feet. Still climbing.

At t = 2: height = -64 + 128 + 80 = 144 feet. Also, maximum height reached. The vertex sits at t = -b/2a = -64/(2(-16)) = 2 seconds.

At t = 3: height = -144 + 192 + 80 = 128 feet. Coming down, passing the same height it had at t = 1.

At t = 4: height = -256 + 256 + 80 = 80 feet. Back at launch height.

At t = 5: height = -400 + 320 + 80 = 0. Impact.

The symmetry is perfect. The factored form (t - 5)(t + 1) makes the roots obvious. Which means the vertex form -16(t - 2)² + 144 makes the maximum obvious. Each form reveals something different.

If you found this helpful, you might also enjoy how many diamonds in a deck of cards or how many pounds is 83 kilograms.

Common Mistakes People Make

Forgetting the negative when factoring out the GCF

If you factor out +16 instead of -16, you get 16(-t² + 4t + 5). Worth adding: the trinomial now has a negative leading coefficient, which makes the next step harder. But always factor out the negative when the leading term is negative. It's not a rule — it's a convenience that prevents errors.

Sign errors in the trinomial

After factoring out -16, the trinomial is t² - 4t - 5. The signs come from division: +64t ÷ -16 = -4t. Consider this: not t² + 4t - 5. Because of that, not t² - 4t + 5. +80 ÷ -16 = -5. Write the division explicitly if you're unsure.

Mixing up the factors of -5

You need multiply to -5, add to -4. Not -5 and -1 (that multiplies to +5). The pair is -5 and +1. That's why the signs matter. If the product is negative, the factors have opposite signs. That said, not -1 and +5 (that adds to +4). If the sum is negative, the larger absolute value is negative.

Forgetting to include the GCF in the final answer

-16(t - 5)(t + 1) is the complete factorization. (t - 5)(t + 1) is incomplete. The -16 is part of the expression. Leaving it off changes the function entirely.

Confusing "factor" with "solve"

Factoring gives you -16(t - 5)(t + 1). Solving gives you t = 5, t = -1. They're related but different tasks. The prompt said "factor this equation" — technically it's an expression, not an equation, until you set it equal to zero. But in practice, people use the terms loosely. Know which output you need.

Alternative Methods (And When to Use Them)

Quadratic formula

t = [-64 ± √(64² - 4(-16)(80))] / (2(-16)) t = [-64 ± √(4096 + 5120)] / -32 t = [-64 ± √9216] / -32 t = [-64 ±

√9216 = 96, so

t = [−64 ± 96] / −32.

Evaluating the two possibilities:

• Using the plus sign: (−64 + 96) = 32 → 32 / −32 = −1.
• Using the minus sign: (−64 − 96) = −160 → −160 / −32 = 5.

Thus the quadratic formula yields the same roots, t = −1 and t = 5, confirming the factored form −16(t − 5)(t + 1).

Completing the Square

Starting from −16t² + 64t + 80, factor out −16: −16(t² − 4t) + 80.
Add and subtract (4/2)² = 4 inside the parentheses: −16[(t² − 4t + 4) − 4] + 80 = −16[(t − 2)² − 4] + 80.
Distribute the −16: −16(t − 2)² + 64 + 80 = −16(t − 2)² + 144.
Setting the expression to zero gives −16(t − 2)² + 144 = 0 → (t − 2)² = 9 → t − 2 = ±3 → t = 5 or t = −1.
This method highlights the vertex (2, 144) directly, which is useful when the maximum height or the axis of symmetry is the primary interest.

Graphical and Technological Approaches

Plotting y = −16t² + 64t + 80 on a calculator or software instantly shows the intercepts at t = −1 and t = 5 and the peak at (2, 144). Numerical solvers (Newton’s method, built‑in root finders) arrive at the same values, especially handy when coefficients are not tidy integers or when dealing with higher‑degree polynomials where factoring by inspection becomes impractical.

When to Choose Each Technique

• Factoring is fastest when the quadratic breaks down into simple integer factors, as it reveals roots and the GCF in one step.
• Quadratic formula works universally; it’s the go‑to when factoring is non‑obvious or when you need an exact expression involving radicals.
• Completing the square excels when you need the vertex form for optimization problems or when deriving the quadratic formula itself.
• Graphing/technology provides a visual check and is invaluable for real‑world data where coefficients are measured rather than exact.

The short version: the expression −16t² + 64t + 80 can be viewed through multiple algebraic lenses—factored form, vertex form, and the quadratic formula—each shedding light on a different facet of the projectile’s motion: the launch and impact times, the peak height, and the symmetry of the trajectory. Mastering when and how to shift between these perspectives turns a seemingly abstract curve into a concrete, interpretable story of motion.