Quadratic Equation Solver guide
How to Solve x² + 5x + 6 = 0
x² + 5x + 6 = 0 has the solutions x = −2 and x = −3. It factors as (x + 2)(x + 3) = 0, so either bracket can be zero. The quadratic formula gives the same answer, since the discriminant is 25 − 24 = 1.
Method 1 — factoring
This equation is a good candidate for factoring, because a = 1 and the numbers are small.
You need two numbers that multiply to give c (6) and add to give b (5).
The pairs multiplying to 6 are 1 and 6, 2 and 3. Of those, 2 and 3 add to 5. That is the pair.
So x² + 5x + 6 = (x + 2)(x + 3).
A product is zero only when one of its factors is zero, so x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.
The sign trap in factoring
The factors are (x + 2) and (x + 3), but the roots are −2 and −3. The signs flip, and this catches people constantly.
The reason is what you do next: to make x + 2 equal zero, x has to be −2.
So the numbers inside the brackets are not the answers. Always take the extra second to solve each bracket rather than reading the roots straight off.
Method 2 — the quadratic formula
With a = 1, b = 5, c = 6:
Discriminant: 5² − 4(1)(6) = 25 − 24 = 1.
√1 = 1, and 2a = 2.
Plus branch: x = (−5 + 1) ÷ 2 = −4 ÷ 2 = −2.
Minus branch: x = (−5 − 1) ÷ 2 = −6 ÷ 2 = −3.
Same answers. The discriminant being a perfect square (1) is exactly why factoring worked so cleanly.
Method 3 — completing the square
Slower here, but worth seeing because it is the method the quadratic formula is derived from.
Start with x² + 5x + 6 = 0 and move the constant: x² + 5x = −6.
Halve the coefficient of x and square it: half of 5 is 2.5, squared is 6.25. Add that to both sides: x² + 5x + 6.25 = 0.25.
The left side is now a perfect square: (x + 2.5)² = 0.25.
Take the square root of both sides: x + 2.5 = ±0.5.
So x = −2.5 + 0.5 = −2, or x = −2.5 − 0.5 = −3.
Checking the answers
For x = −2: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0. Correct.
For x = −3: (−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0. Correct.
Both work, as they must — three methods agreeing and a substitution check is about as certain as arithmetic gets.
What the graph looks like
The parabola crosses the x-axis at −2 and −3.
Its vertex sits halfway between them, at x = −2.5. Putting that back into the equation gives y = −0.25, so the lowest point is (−2.5, −0.25).
Because a = 1 is positive, the parabola opens upward, so that vertex is a minimum.
The calculator on this page gives you the vertex and the factored form alongside the roots, which is usually what a question asks for next.
