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Algebra 1 9-12 / Week 10 / Monday
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Week 10 · Quadratic Equations

Monday

The landing time
// When is the height zero?
⏱ about 20 min

Monday: The Landing Time

The bottle rocket sits on its launcher in the backyard, fins taped on, a grown-up watching from the steps.

Comet pumps six times and steps back. "Everyone clear?" The rocket launches, climbs, hangs for a moment and lands in the grass.

Wren clicks the stopwatch. "Four seconds. What do you notice about our height rule?"

He writes it on the whiteboard: h(t) = -5t² + 20t. "Our model from the stopwatch data. Height in meters, t in seconds."

"So when is the height zero?" Comet asks. "That is the landing."

Nova hovers over the equation, her light on the t. "Would you like a hint? Both terms share a factor."

"-5t times (t - 4)," Comet says. "Zero when t is 0 or t is 4. Launch and landing!"

"Two solutions," Wren says. "One of them is the answer we want."

Comet, Wren and Nova in the backyard as the bottle rocket launches, with a grown-up watching from the steps.

A quadratic equation

The crew's height rule is h(t) = -5t² + 20t. Setting the height to zero gives -5t² + 20t = 0.

An equation with a squared letter, and no higher power, is a quadratic equation. Its standard form is ax² + bx + c = 0.

Here a = -5, b = 20 and c = 0. The numbers come from the crew's own stopwatch timings, not from a law.

This week the crew learns four ways to solve any quadratic equation. Today: factoring.

A solved problem to study

  1. Start with -5t² + 20t = 0.
  2. Factor out the common factor: -5t(t - 4) = 0.
  3. Use the zero-product rule: if two factors multiply to 0, one of them is 0.
  4. So -5t = 0 or t - 4 = 0.
  5. Solve each: t = 0 or t = 4.
  6. Check: put 4 back in. -5 × 4² + 20 × 4 = 0. True.

Why does the rule work? A product is zero only when a factor is zero. No other pair of numbers multiplies to 0.

So every quadratic you can factor turns into two small linear equations. You solved those in week 2.

t = 0 is the launch. t = 4 is the landing. Both are true, and the story picks the one you need.

SOLVE BY FACTORING
  • Read the question.
  • Tap your answer.
Solve -5t² + 20t = 0 by factoring. What are the solutions?
What are the solutions of x² - 6x + 5 = 0?
Use the zero-product rule on (x - 3)(x + 2) = 0. What are the solutions?
Which is the factored form of x² - 6x + 5?
The crew's rule is h(t) = -5t² + 20t. How many seconds after launch does the rocket land? Type the number.
WHY THIS EXERCISESetting the height to zero and solving finds the landing time. Factoring turns one quadratic into two linear equations.
StatementTrue or false?
If two factors multiply to 0, at least one of them is 0.?
-5 × 4 × 4 + 20 × 4 = 0?
-5 × 2 × 2 + 20 × 2 = 0?
The equation -5t² + 20t = 0 has exactly one solution.?
x² - 6x + 5 = 0 is a quadratic equation.?
WHY THIS EXERCISEThe zero-product rule only works with a zero on one side. Checking by substitution proves a solution.
Try it
Write (x - 2)(x + 5) = 0 on an index card. Solve it with the zero-product rule, then multiply the factors back out.
Now write x² + 3x - 10 = 0 on another card. Factor it and compare with the first card.
Draw the rocket's flight as a curve from t = 0 to t = 4. Mark the two times the height is zero.

Strong start. Tomorrow you meet three more ways to solve a quadratic and compare them side by side.