Concepts · 3 min read
The Quadratic Formula: Where It Comes From
Start with the general equation
Begin with ax² + bx + c = 0 and divide by a: x² + (b/a)x + c/a = 0. Move the constant across: x² + (b/a)x = −c/a.
Complete the square
Add (b/2a)² to both sides so the left becomes a perfect square: (x + b/2a)² = (b² − 4ac) / 4a². Take square roots: x + b/2a = ±√(b² − 4ac) / 2a. Subtract b/2a and you have the formula.
What the discriminant tells you before you finish
The expression under the root, D = b² − 4ac, is the discriminant. Positive means two real roots, zero means one repeated root, and negative means two complex roots.
A worked example with the solver
- Factored form: (x − 4)(x + 1)
- Discriminant D = 25
- Vertex: (1.5, −6.25)
Step-by-step working 6 steps
- 1Identify a, b and c
x² − 3x − 4 = 0 a = 1, b = −3, c = −4
- 2Calculate the discriminant
D = b² − 4ac = (−3)² − 4(1)(−4) = 25
D is positive: two different real roots.
- 3Apply the quadratic formula
x = (−b ± √D) / 2a = (3 ± √25) / 2 = (3 ± 5) / 2
- 4Solve for both roots
x₁ = 4 x₂ = −1
- 5Check by substitution
f(4) = 0 f(−1) = 0
Both should be 0 (allowing for tiny rounding).
- 6Vertex of the parabola
x = −b / 2a = 1.5, y = −6.25 → vertex (1.5, −6.25)
The parabola opens upward, so the vertex is the minimum point.
Frequently asked questions
Why does the formula have a ±?
Because squaring loses the sign: both a positive and a negative number have the same square, so there are two possible roots.
Is there a formula for cubic equations?
Yes, but it is much longer; for degree five and above there is no general formula using ordinary roots.
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