Week 7 — Two Answers, One Setting: A Short Quadratic Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

Assignment 7 of the term — solve four short quadratic equations, explain what each answer means, and tell someone new to algebra why an equation can give an answer the real situation throws out.

This week you learned to put a quadratic equation in standard form, to solve it by factoring and the zero-product property without dividing by the variable, to use the square root property, to complete the square, to use the quadratic formula, and to read the discriminant. In Week 6 you factored polynomials such as x² − 6x + 5 = (x − 1)(x − 5); this week you set expressions like that equal to 0 and solve them. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied algebra why an equation can give an answer the real situation throws out. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

The task

You will use four equations from this week: 3x² − 12x = 0, (x − 3)² = 16, the equation x² + 6x + 5 = 0, and a patio problem. The patio is rectangular, 3 feet longer than it is wide, and covers 40 square feet, so its width w satisfies w(w + 3) = 40, which is w² + 3w − 40 = 0 in standard form.

Work these four problems, showing your arithmetic:

1. Solve 3x² − 12x = 0 by factoring. What is its nonzero solution?
2. Solve (x − 3)² = 16 with the square root property. What is the smaller solution?
3. Completing the square turns x² + 6x + 5 = 0 into (x + 3)² = k. What is k?
4. A rectangular patio is 3 feet longer than it is wide, and its area is 40 square feet. Its width w satisfies w² + 3w − 40 = 0. What is its length, in feet?

Beside each answer, add one brief interpretation: what that value shows about the equation and one use or limit of it. For problem 1, name both solutions and say what dividing both sides by x would lose. For problem 2, say why there are two solutions and check both. For problem 3, say why the same number goes on both sides, then finish the solve and check it. For problem 4, say which solution of the width equation the setting rejects and why, and check the area.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

Explain it to a non-expert

A neighbor is helping you lay out the patio and has never studied algebra. They saw your equation w² + 3w − 40 = 0 give two answers and are confused. Write a 3–5 sentence message for your neighbor: why a squared equation can give two answers, why only one of them can be the patio's width, and one honest limit of the equation, such as what it assumes about the patio's shape. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine a square picture frame that must enclose 64 square inches of photo, so its side s satisfies s² = 64. Model answer: "The equation asks for a number that times itself makes 64. Both 8 and negative 8 do that, because a negative times a negative is positive, so the math gives two answers. A side of a frame cannot be negative, so the frame's side is 8 inches. The equation assumes a perfect square opening, so a frame with a mat border would need a different equation."

Use that pattern for the patio, with your own explanation.

Deliverables

Submit a single quadratic report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows about the equation and one use or limit of it;
• your message for your neighbor, explaining why the equation gave two answers, why only one can be the patio's width, and one honest limit; and
• a one-line restatement of the equations you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch with a short note. Every form is graded by the same scoring criteria. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your answers. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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