💡 What You Will Learn
Factoring a quadratic like x² + bx + c means rewriting it as (x + p)(x + q). The key is finding two numbers p and q that multiply to c (the constant) and add up to b (the middle coefficient). Once you see the pattern, it clicks!
A quadratic expression x² + bx + c can be rewritten as a product of two binomials: (x + p)(x + q).
The relationship: p · q = c (they multiply to the constant c) and p + q = b (they add to the middle coefficient b).
(x + 5)(x − 1)
= x² − x + 5x − 5
= x² + 4x − 5
x² + 4x − 5
Find: p·q=−5 and p+q=4
= (x + 5)(x − 1) ✅
In x² + 4x − 5, b = 4. Your two numbers must add to this.
In x² + 4x − 5, c = −5. Your two numbers must multiply to this.
When you expand (x + p)(x + q) using FOIL:
x² + qx + px + pq = x² + (p+q)x + (p·q)
So b = p+q and c = p·q — that's where our two rules come from!
In x² + bx + c, b is the middle coefficient, c is the last number.
Example: x² + 4x − 5 → b = 4, c = −5
List all pairs of integers that multiply to c (including negatives!).
For c = −5: (1, −5), (−1, 5), (5, −1), (−5, 1)
Which pair sums to b = 4?
5 + (−1) = 4 ✅ → p = 5, q = −1
(x + p)(x + q) = (x + 5)(x − 1) ✅
Note: x + (−1) is written as x − 1.
| Step | What To Do | For x² + 4x − 5 |
|---|---|---|
| 1 | Read b and c | b=4, c=−5 |
| 2 | Factor pairs of c | (1,−5),(−1,5),(5,−1) |
| 3 | Which adds to b? | 5+(−1)=4 ✅ |
| 4 | Write answer | (x+5)(x−1) |
If b is also positive → both p and q are positive
If b is negative → both p and q are negative
Examples: x²+5x+6=(x+2)(x+3) x²−5x+6=(x−2)(x−3)
One factor is positive, one is negative. The larger absolute value takes the sign of b.
Examples: x²+4x−5=(x+5)(x−1) x²−4x−5=(x−5)(x+1)
Both factors are positive.
x²+7x+12=(x+3)(x+4)
3×4=12 ✅ 3+4=7 ✅
Both factors are negative.
x²−7x+12=(x−3)(x−4)
(−3)(−4)=12 ✅ −3+(−4)=−7 ✅
Bigger factor is positive.
x²+4x−5=(x+5)(x−1)
5×(−1)=−5 ✅ 5+(−1)=4 ✅
Bigger factor is negative.
x²−4x−5=(x−5)(x+1)
(−5)×1=−5 ✅ −5+1=−4 ✅
| c sign | b sign | Factor signs | Example |
|---|---|---|---|
| + positive | + positive | (x+p)(x+q) | x²+7x+12=(x+3)(x+4) |
| + positive | − negative | (x−p)(x−q) | x²−7x+12=(x−3)(x−4) |
| − negative | + positive | (x+big)(x−small) | x²+4x−5=(x+5)(x−1) |
| − negative | − negative | (x−big)(x+small) | x²−4x−5=(x−5)(x+1) |
When c = (b/2)², both factors are identical!
x² + 4x + 4 → (b/2)² = 4 ✅ → (x + 2)(x + 2) = (x + 2)²
FOIL stands for First, Outer, Inner, Last.
(x + p)(x + q) = x² + qx + px + pq = x² + (p+q)x + pq
Just verify p + q = b and p × q = c.
For (x+5)(x−1): 5+(−1)=4=b ✅ 5×(−1)=−5=c ✅
If your check fails, flip the sign of one or both factors and try again. Signs are the #1 error in factoring!