🔍 Grade 10 · Lesson 1 of 12
Factorization Foundry: Factoring Quadratics with the Area Model
When the leading coefficient isn't 1, a rectangle built from tiles shows you exactly how the trinomial splits into two factors.
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The big idea: A quadratic like ax² + bx + c can be pictured as the area of a rectangle. If you split the middle term bx into two pieces that let you group the rectangle into two smaller rectangles sharing a common side, that shared side and the other two sides are exactly the factors of the quadratic. The trick is finding the right split: two numbers that multiply to a×c and add to b.
Why the leading coefficient makes it harder
Factoring x² + bx + c is friendly: you just find two numbers that multiply to c and add to b. But the moment the leading coefficient a is not 1 — say 2x² + 7x + 3 — that shortcut breaks. The x² term can come from more than one pair of factors, so you need a more systematic way in.
The area model gives you that system: build the trinomial as the area of a rectangle, and read the factors straight off its side lengths.
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The ac method
To factor ax² + bx + c, find two numbers that multiply to a×c and add to b. Use them to split the middle term bx into two pieces, then factor the four resulting terms by grouping.
\[ \text{find } m, n \text{ such that } m \times n = ac \quad \text{and} \quad m + n = b \]
Once you have m and n, rewrite bx as mx + nx and the quadratic becomes four terms you can group in pairs.
🎮 Factorization Foundry LIVE
Set the two binomial factors (px + q)(rx + s); the box fills with the four partial products and their sum is the trinomial. Positive terms are green, negative red. Try (2x + 1)(x + 3) and (3x + 1)(x − 2) from the examples below.
Building the rectangle
Picture the trinomial as tiles: one large a×x² tile, some x-strip tiles worth 1×x each, and c unit tiles. Arranging them into a single rectangle forces the strip tiles to split into two groups — one group along each dimension. Group the four pieces in pairs and pull out the common factor from each pair; both pairs will share the same leftover binomial, which becomes one side of the rectangle.
\[ ax^2 + bx + c \;\longrightarrow\; (px + q)(rx + s) \]
The two side lengths of the completed rectangle are the two binomial factors.
📝 Worked example: Factor 2x² + 7x + 3.
- ac = 2 × 3 = 6, and b = 7. Find two numbers that multiply to 6 and add to 7: 6 and 1.
- Split the middle term: 2x² + 6x + x + 3.
- Group and factor each pair: 2x(x + 3) + 1(x + 3).
- Both groups share the factor (x + 3), so pull it out: (2x + 1)(x + 3).
✓ 2x² + 7x + 3 = <strong>(2x + 1)(x + 3)</strong>. Check: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. ✓
📝 Worked example: Factor 3x² − 5x − 2.
- ac = 3 × (−2) = −6, and b = −5. Find two numbers that multiply to −6 and add to −5: −6 and 1.
- Split the middle term: 3x² − 6x + x − 2.
- Group and factor each pair: 3x(x − 2) + 1(x − 2).
- Pull out the shared factor (x − 2): (3x + 1)(x − 2).
✓ 3x² − 5x − 2 = <strong>(3x + 1)(x − 2)</strong>. Check: (3x + 1)(x − 2) = 3x² − 6x + x − 2 = 3x² − 5x − 2. ✓
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Check for a GCF first, and watch your signs
Always pull out a greatest common factor from all three terms before starting the ac method — it makes the numbers smaller and easier to work with. And when ac is negative, one of your two numbers m and n will be negative and the other positive; when ac is positive but b is negative, both m and n are negative.
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A repeated factor means a repeated root
If the two binomial factors turn out identical — like (2x + 1)(2x + 1) for 4x² + 4x + 1 — the quadratic is a perfect square trinomial, and its graph touches the x-axis at just one repeated point instead of crossing it at two.
Check your understanding
1. To factor 2x² + 7x + 3 by the ac method, which pair of numbers do you need?
ac = 2 × 3 = 6. The two numbers must multiply to 6 and add to 7, so 6 and 1 work (6 × 1 = 6, 6 + 1 = 7).
2. Factor 3x² − 5x − 2 completely.
Splitting −5x as −6x + x gives 3x(x−2) + 1(x−2) = (3x + 1)(x − 2).
3. In the area model, what do the side lengths of the completed rectangle represent?
The rectangle's area is the trinomial, so its two side lengths are exactly the two factors that multiply to give it.
4. Why do we split the middle term bx into two pieces when factoring ax² + bx + c?
Splitting bx lets you factor by grouping: each pair of terms shares a common binomial, and that shared binomial becomes one side of the rectangle.
5. Factoring 4x² + 4x + 1 gives (2x + 1)(2x + 1) = (2x + 1)². What does this tell you about the quadratic?
When both factors are identical, the quadratic is a perfect square trinomial, and its graph touches the x-axis at a single repeated root instead of crossing at two.
✅ Key takeaways
- The area model treats ax² + bx + c as the area of a rectangle whose side lengths are its factors.
- The ac method: find two numbers that multiply to a×c and add to b, then split the middle term.
- After splitting, factor by grouping — pull the common binomial out of each pair of terms.
- Always check for a greatest common factor first, and track signs carefully when ac is negative.
- Always verify by multiplying your two factors back out — it should reproduce the original trinomial.
- Identical factors, like (2x + 1)(2x + 1), signal a perfect square trinomial with a repeated root.