Quadratic Equation Calculator

Solve ax² + bx + c = 0 and inspect the parabola.

Enter coefficients

D > 02 real roots

D = 01 repeated root

D < 02 complex roots

Positive discriminant: the graph crosses the x-axis twice.

Discriminant1

Vertex(1.5, -0.25)

Axisx = 1.5

OpeningUpward

Y-intercept2

Root checkΣ 3; Π 2

Standard formy = x² - 3x + 2

Best for reading a, b, and c.

Vertex formy = (x - 1.5)² - 0.25

Reveals the turning point and axis.

Factored formy = (x - 1)(x - 2)

Connects the equation to its real or complex roots.

Write the equation in standard form: x² - 3x + 2 = 0.
Calculate the discriminant: D = b² - 4ac = (-3)² - 4(1)(2) = 1.
Substitute into x = (-b ± √D) / (2a) to obtain x₁ = 1 and x₂ = 2.
Find the vertex: h = -b/(2a) = 1.5, then k = f(h) = -0.25.
The axis of symmetry is x = 1.5; the parabola opens upward because a is positive.

The curve shows y = ax² + bx + c. Move the explorer to see how an x-value maps to the curve.

Explore x = 1.5f(1.5)-0.25

Graph point details
Feature	Coordinate or value	Meaning
Vertex	(1.5, -0.25)	Minimum point
Axis	x = 1.5	Vertical line of symmetry
Y-intercept	(0, 2)	Where the graph crosses the y-axis
Roots	1, 2	X-intercepts

Explore advanced parabola geometry

These values describe the complete graph, not only its roots.

Turning pointMinimum at (1.5, -0.25)

DomainAll real numbers

Rangey ≥ -0.25

Focus(1.5, 0)

Directrixy = -0.5

Focal length0.25

Root separation1

Suggested methodQuadratic formula; factoring may also be possible

Results are displayed in 2-3 decimal places only. Metric and Imperial conversion is not applicable to unitless coefficients.