Mathematics › 🌆 Grade 9 › Rectangle Resizer: Factoring Quadratics with Area Models

Rectangle Resizer: Factoring Quadratics with Area Models

A trinomial is just the area of a rectangle waiting to be rebuilt — find its side lengths and you have factored it.

Grade 9Algebra 1

Rectangle Resizer: Factoring Quadratics with Area Models — illustration

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The big idea: Multiplying two binomials, like (x + 3)(x + 4), builds a rectangle whose area is a trinomial. Factoring runs that process backward: given the trinomial, you rearrange its area tiles into a rectangle and read the side lengths off its edges. For x² + bx + c, those side lengths are two numbers that multiply to c and add to b.

🎯 By the end, you'll be able to

Represent a trinomial as the area of a rectangle built from algebra tiles
Factor x² + bx + c by finding two numbers that multiply to c and add to b
Verify a factoring result by expanding it back out
Solve a simple quadratic equation by factoring and applying the zero product property

📎 You should already know

Multiplying binomials (FOIL)
Integer multiplication and addition, including negatives

Area, built and rebuilt

When you multiply two binomials, like (x + 3)(x + 4), you are really finding the area of a rectangle with those side lengths: one big x-by-x square, some x-by-1 strips, and some 1-by-1 unit tiles. Multiply it out and you get a trinomial: x² + 7x + 12.

Factoring is the reverse trip: starting from the trinomial, rearrange its tiles back into a clean rectangle, and read the two side lengths off its edges. Those side lengths are the factors.

🔑 Factoring reverses expanding

To factor x² + bx + c into (x + p)(x + q), you need two numbers p and q whose product is c and whose sum is b. Find that pair, and the rectangle snaps together.

\[ (x+p)(x+q) = x^2 + (p+q)x + pq \]

Expanding shows exactly where b and c come from: b = p + q, and c = p × q. Factoring works backward from b and c to find p and q.

🎮 Rectangle Resizer LIVE

Rearrange algebra tiles into a rectangle; its side lengths are the factors of the quadratic.

The two-number trick

To factor x² + bx + c, list pairs of numbers that multiply to c, then check which pair also adds to b. Once found, those two numbers p and q go straight into (x + p)(x + q).

📝 Worked example: Factor x² + 7x + 12.

You need two numbers that multiply to 12 and add to 7.
Try 3 and 4: 3 × 4 = 12, and 3 + 4 = 7. That's the pair.
Write the factors using those numbers.

✓ x² + 7x + 12 = <strong>(x + 3)(x + 4)</strong>.

📝 Worked example: Factor x² − 2x − 15.

You need two numbers that multiply to −15 and add to −2.
Try −5 and 3: (−5) × 3 = −15, and (−5) + 3 = −2. That's the pair.
Write the factors using those numbers.

✓ x² − 2x − 15 = <strong>(x − 5)(x + 3)</strong>.

✨ Let the signs guide your search

If c is positive and b is positive, both numbers are positive. If c is positive and b is negative, both numbers are negative. If c is negative, one number is positive and one is negative — and the larger-magnitude one takes the sign of b.

⚠️ Always check by expanding back out

After factoring, multiply your two binomials back together (FOIL) and confirm you land back on the original trinomial. This catches sign mistakes before they turn into a wrong final answer.

From factoring to solving

Once a quadratic is factored and set equal to 0, the zero product property says that if two factors multiply to 0, at least one of them must itself be 0. So if (x + 3)(x + 4) = 0, then either x + 3 = 0 or x + 4 = 0 — giving x = −3 or x = −4.

Check your understanding

1. Factor x² + 5x + 6.

2 × 3 = 6 and 2 + 3 = 5, so x² + 5x + 6 = (x + 2)(x + 3).

2. Factor x² − x − 6.

−3 × 2 = −6 and −3 + 2 = −1, so x² − x − 6 = (x − 3)(x + 2).

3. Which pair of numbers multiplies to −12 and adds to 1, factoring x² + x − 12?

4 × (−3) = −12 and 4 + (−3) = 1, matching the coefficients of x² + x − 12.

4. Solve (x − 5)(x + 2) = 0.

By the zero product property, x − 5 = 0 gives x = 5, and x + 2 = 0 gives x = −2.

5. In the rectangle area model for factoring, what do the two side lengths of the rectangle represent?

The rectangle's width and height are exactly the two binomial factors whose product gives the trinomial's area.

✅ Key takeaways

Multiplying two binomials builds a rectangle; its area is a trinomial — factoring runs that process in reverse.
To factor x² + bx + c, find two numbers that multiply to c and add to b.
The signs of b and c tell you what signs to look for in the two numbers.
Always check a factoring result by expanding it back out (FOIL) to confirm it matches the original trinomial.
Once a quadratic is factored and set to 0, the zero product property lets you solve it directly.