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Grade 9 · Explain a binomial area product · 694 of 750

Explain a binomial area product · practice 75

Evaluate (x + 3)(x + 7) at x = 3 by finding all four rectangle-area parts. Follow the calculation, then test the quantities in the animated example.

Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.

01 · Make a prediction

Your practice question

Evaluate (x + 3)(x + 7) at x = 3 by finding all four rectangle-area parts.

Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.

02 · Explore the model

See the mathematical relationship move.

Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.

Watch the relationship

Paused
Explain a binomial area product · practice 75. Square x²: 9. Two strips x(a+b): 30. Corner ab: 21. Whole product: 60One rectangle, four area productsx² = 9ax = 9bx = 21ab = 21Sides: (3+3) and (3+7)Sum of four areas = 60
x, a, and b are positive in this area model; signed algebra requires additional interpretation. The picture is fitted with one common scale for both dimensions. All area terms are in square units.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Square x²
9
Two strips x(a+b)
30
Corner ab
21
Whole product
60

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

The mathematical idea

Partitioning a rectangle expresses one product as the sum of four areas. The two cross terms combine into x(a+b), giving the coefficient of x. Reversing that comparison explains factoring a monic quadratic when its positive constants have the required sum and product. Starting quantities: Added length a = 3; Added width b = 7; Shared length x = 3.

(x+a)(x+b) = x²+(a+b)x+ab

03 · Reflect and transfer

Explain what changes and why.

Which two regions explain the middle coefficient in the expanded quadratic?

This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.