Question
Solve the quadratic equation: x² + 5x + 6 = 0
Solution — Step by Step
This equation is perfectly set up for factoring. Here's why we try factoring first: it's the fastest method when the discriminant is a perfect square, and a quick mental check confirms that before we even write anything down.
Step 1: Check if the equation is in standard form.
It is. a = 1, b = 5, c = 6. No rearranging needed.
Step 2: Find two numbers that multiply to ac and add to b.
We need two numbers such that:
- Product = a × c = 1 × 6 = 6
- Sum = b = 5
Think: what pairs multiply to 6? Options are (1, 6), (2, 3), (−1, −6), (−2, −3). Which pair adds to 5? That's 2 and 3.
Step 3: Rewrite the middle term using these two numbers.
x² + 2x + 3x + 6 = 0
We split 5x into 2x + 3x. This doesn't change the equation — we're just rewriting the same thing.
Step 4: Factor by grouping.
x(x + 2) + 3(x + 2) = 0
Both groups share the factor (x + 2):
(x + 2)(x + 3) = 0
Step 5: Apply the zero product property.
If the product of two factors is zero, at least one factor must be zero.
x + 2 = 0 → x = −2
x + 3 = 0 → x = −3
Verification:
- x = −2: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0 ✓
- x = −3: (−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0 ✓
Answer
x = −2 or x = −3
Why This Works
The factoring method relies on reversing the FOIL (or expansion) process. When we multiply (x + 2)(x + 3), we get x² + 3x + 2x + 6 = x² + 5x + 6. So these factors are exactly the reverse of the original equation.
The zero product property — if ab = 0 then a = 0 or b = 0 — is a fundamental property of real numbers. Integers and fractions have this property; it's what makes factoring work as a solving technique.
Alternative Method: Quadratic Formula
If factoring didn't come to mind, the formula gives the same answer:
a = 1, b = 5, c = 6
D = b² − 4ac = 25 − 24 = 1
x = (−5 ± √1) / 2 = (−5 ± 1) / 2
x = (−5 + 1)/2 = −4/2 = −2
x = (−5 − 1)/2 = −6/2 = −3
Same roots, more steps. Factoring was faster here.
💡 Expert Tip
When a = 1, mentally test pairs that multiply to c and add to b. This takes under 10 seconds for simple cases. If no integer pair works in 10 seconds, move to the formula — don't force factoring.
Common Mistake
⚠️ Common Mistake
Don't write x = 2 or x = 3 (positive values).
The equation is (x + 2)(x + 3) = 0. Setting x + 2 = 0 gives x = −2, not x = 2. The sign flips. This is the most common error on this type of question — students find the right factors but forget the negative sign in the roots.