Solve real-world problems modeled by quadratic equations, including projectile motion and area, and interpret the reasonableness of solutions in context (TEKS A.8A, A.6B).

What this topic is asking

TEKS A.8A and A.6B ask you to apply quadratics to real situations, projectile motion, area, and revenue, and to interpret the solutions, including rejecting answers that make no physical sense. These are often the higher-tariff, multi-step items in the Quadratic Functions and Equations category, where credit is spread across setting up the model, solving, and interpreting.

Projectile motion

A projectile's height over time is a downward-opening parabola, commonly $h(t) = -16t^2 + v_0 t + h_0$ in feet and seconds (the $-16$ is half the acceleration of gravity). Two questions recur:

• When does it hit the ground? Set $h = 0$ and solve. One root is usually the launch ($t = 0$ if $h_0 = 0$); the other is the landing.
• What is the maximum height? The peak is the vertex. Find the time $t = \frac{-b}{2a}$, then substitute to get the height.

Area and other models

For an area problem, express the area as a product of dimensions and set it equal to the given value, producing a quadratic. A garden 3 feet longer than wide with area 40 gives $w(w + 3) = 40$, so $w^2 + 3w - 40 = 0$. Solve, then reject the negative root because a length cannot be negative.

Revenue problems behave similarly: revenue is often price times quantity where one depends on the other, giving a parabola whose vertex is the maximum revenue.

Interpreting and rejecting solutions

The interpretive step is what these items reward. A quadratic usually has two solutions, but the context often makes one impossible:

• Negative time before launch is rejected.
• Negative length, width, or count is rejected.
• The launch root ($t = 0$) is not the "landing" answer.

Always finish by stating the meaningful solution with units and, where asked, why the other root is discarded.

How STAAR examines this topic

• Multiple choice. Find a landing time or maximum height; the "vertex instead of zero" or "wrong root" answer is the standard distractor.
• Equation editor and number entry. Set up the model, solve, and report the realistic value.
• Multi-step items. Write the equation, solve, and interpret, with credit across the steps.

A clarifying idea is that the vertex and the zeros answer different questions: the zeros are when the quantity is zero (ground level, break-even), while the vertex is the extreme value (highest point, maximum revenue). Reading which one the prompt wants is the difference between two plausible-looking answers, and the words "how high", "maximum", or "least" point to the vertex while "when does it land", "hits the ground", or "break even" point to a zero.

Try this

Q1. A ball's height is $h(t) = -16t^2 + 32t$. When does it land? [2 points]

• Cue. $-16t(t - 2) = 0$, so $t = 0$ or $t = 2$; it lands at $t = 2$ s.

Q2. A rectangle is 2 cm longer than wide with area 24. Find the width. [2 points]

• Cue. $w(w + 2) = 24 \Rightarrow w^2 + 2w - 24 = 0 \Rightarrow (w + 6)(w - 4) = 0$; width 4 cm.