What is the discriminant?

The discriminant D = b²-4ac determines the nature of roots. D>0: two real roots, D=0: one repeated root, D<0: two complex roots.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. x = -b/(2a), y = f(-b/(2a)). If a>0, it's a minimum; if a<0, it's a maximum.

Quadratic Equation Solver - Find Roots of ax² + bx + c = 0

The quadratic formula, and what the discriminant tells you

For ax² + bx + c = 0 the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is the discriminant, and its sign alone tells you whether you get two real roots, one repeated root, or a complex pair — before you finish the arithmetic.

What the discriminant decides

Discriminant D = b² − 4ac	Roots	Graph of y = ax² + bx + c
D > 0, a perfect square	Two distinct rational roots	Crosses the x-axis twice at nice values
D > 0, not a perfect square	Two distinct irrational roots	Crosses the x-axis twice
D = 0	One repeated real root, x = −b/2a	Touches the x-axis at the vertex
D < 0	Two complex conjugate roots	Never touches the x-axis

Three worked examples

D > 0 — x² − 5x + 6 = 0

a = 1, b = −5, c = 6

D = 25 − 24 = 1

x = (5 ± 1) / 2

x = 3 and x = 2

D = 0 — x² + 6x + 9 = 0

a = 1, b = 6, c = 9

D = 36 − 36 = 0

x = −6 / 2

x = −3, a repeated root; the parabola just touches the axis

D < 0 — x² + x + 1 = 0

a = 1, b = 1, c = 1

D = 1 − 4 = −3

x = (−1 ± i√3) / 2

x ≈ −0.5 + 0.866i and −0.5 − 0.866i

Everything else you can read off the coefficients

Sum of roots = −b / a
Product of roots = c / a
Vertex is at x = −b / 2a, and the axis of symmetry is that vertical line
Minimum or maximum value = c − b²/4a, a minimum when a > 0 and a maximum when a < 0
Y-intercept = c, always

When factoring beats the formula

If the discriminant is a perfect square, the equation factors over the integers and factoring is faster. For x² − 5x + 6 you need two numbers multiplying to 6 and adding to −5 — −2 and −3 — so it is (x − 2)(x − 3) on sight. When a ≠ 1, look for factors of a×c that add to b. The formula never fails, but it is slower, and on a written exam it is where sign slips creep in.

Completing the square, which is where the formula comes from

Divide through by a: x² + (b/a)x + c/a = 0.
Move the constant: x² + (b/a)x = −c/a.
Add (b/2a)² to both sides to make the left side a perfect square.
It becomes (x + b/2a)² = (b² − 4ac) / 4a².
Take the square root of both sides and subtract b/2a. That is the quadratic formula.

Quadratic Equation Solver

Solution

Related Tools

The Quadratic Formula

The quadratic formula, and what the discriminant tells you

What the discriminant decides

Three worked examples

Everything else you can read off the coefficients

When factoring beats the formula

Completing the square, which is where the formula comes from

More questions people ask

Frequently Asked Questions

Answers, not just a number

Want this calculator on your own site?