Quadratic equations explained
Three reliable ways to solve a quadratic, and how the discriminant tells you which way is best.
What is a quadratic equation?
A quadratic equation is any equation you can rearrange into the form ax² + bx + c = 0 with a ≠ 0. The shape on the page is a parabola, and the values of x that make the equation true are the points where that parabola crosses the x-axis.
Three solution methods
Method 1: Factoring
The fastest method when it works. If you can write ax² + bx + c as a product (px + q)(rx + s), then by the zero-product property at least one factor must be zero. So x = −q/p or x = −s/r. Take x² − 5x + 6 = 0: it factors as (x − 2)(x − 3) = 0, giving x = 2 or x = 3 in three lines flat.
Factoring works only when the roots are rational. If the discriminant is not a perfect square, you will not find integer factors and you should switch methods.
Method 2: Completing the square
Add and subtract the right constant to turn x² + bx into a perfect square. Specifically, x² + bx = (x + b/2)² − (b/2)². The method always works and is the proof behind the quadratic formula. Worked example for x² − 6x + 5 = 0:
- Move the constant:
x² − 6x = −5. - Add
(−6/2)² = 9to both sides:x² − 6x + 9 = 4. - Recognise the left side as a square:
(x − 3)² = 4. - Take the square root:
x − 3 = ±2, so x = 1 or x = 5.
Method 3: The quadratic formula
This always works, and you should memorise it: x = (−b ± √(b² − 4ac)) / (2a). Plug in a, b, c and read off the two roots. The plus gives one, the minus gives the other.
The discriminant
The discriminant D = b² − 4ac is the most informative single number in elementary algebra. Three cases:
- D > 0 — two distinct real roots. The parabola crosses the x-axis at two points.
- D = 0 — one repeated real root. The parabola just touches the x-axis at its vertex.
- D < 0 — two complex-conjugate roots. The parabola sits entirely above (if a > 0) or below the x-axis.
Vertex, axis of symmetry, and the parabola
Every quadratic y = ax² + bx + c traces a parabola. The vertex is at x = −b/(2a), and that is also the equation of the axis of symmetry. Substituting that x back into the equation gives the y-coordinate of the vertex — the minimum if a > 0, the maximum if a < 0. This is the geometric story behind the algebraic discriminant: a positive discriminant means the parabola dips through the x-axis; a zero discriminant means the vertex sits on the axis; a negative discriminant means the parabola never touches it.
Vieta's formulas
For roots r₁ and r₂ of ax² + bx + c = 0, the sum r₁ + r₂ equals −b/a and the product r₁ · r₂ equals c/a. They give you a free sanity check on any quadratic solution: add and multiply the roots, see if the answers match. They also let you build a quadratic with the roots you want — if you want roots 4 and −1, the equation is x² − 3x − 4 = 0.
When to use which method
- Try factoring first when the coefficients are small integers — it is the fastest.
- Use the formula when factoring fails or the coefficients are awkward.
- Complete the square when you need the vertex form or are deriving anything theoretically.
If you would rather just punch in a, b, c and get the answer instantly, the quadratic solver shows the discriminant, vertex, axis of symmetry, and a graph alongside the roots.