Quadratic Formula Calculator

Solve quadratic equations in standard form with ease using our Quadratic Formula Calculator. Instantly find roots (solutions), vertex, discriminant, and graph for any quadratic equation. Perfect for students, teachers, and engineers who need quick, accurate results.

Quadratic Calculator

Standard Form (ax² + bx + c)

Solutions:

Vertex Form:

How It Works

1

Enter Values

Enter the coefficients a, b, and c for the quadratic equation ax² + bx + c = 0.

2

Calculate Instantly

Click Calculate to get:

  • The roots (x-intercepts)
  • Vertex form of the equation
  • Discriminant value to identify root types (real, repeated, or complex)
3

Visualize the Graph

See your quadratic equation plotted automatically. The graph displays the parabola, vertex, and points of intersection with the x-axis.

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Understanding Quadratic Calculator

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable (x), expressed as:

  • [ ax^2 + bx + c = 0 ]

    • Where:

  • a ≠ 0 → coefficient of x² (controls parabola shape)
  • b → coefficient of x (affects symmetry and direction)
  • Mixed Numbers: Combine whole and fractional parts (e.g., 2 1/2)
  • c → constant term (y-intercept)

Quadratic Formula

The quadratic formula finds the roots of the equation using:

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

The term b² - 4ac is called the discriminant (D):

D > 0 → Two distinct real roots

D = 0 → One real (repeated) root

D < 0 → Two complex roots

Quadratic Formula

Given:

a = 1, b = -3, c = -4

[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)} = \frac{3 \pm \sqrt{25}}{2} ]
The term b² - 4ac is called the discriminant (D):

Roots: x₁ = 4, x₂ = -1

Vertex Form: y = (x - 1.5)² - 6.25

Vertex Form Formula

A quadratic function can also be expressed as:

[ y = a(x - h)^2 + k ]

Where:
  • • (h, k) = vertex of the parabola
  • • = -b / 2a
  • • k = f(h)(substitute h into the equation)

Example:
  • • For a = 1, b = -4, c = 3
  • • h = 2 → k = -1
  • • Vertex: (2, -1)

Applications of Quadratic Equations

Quadratic equations are widely used in:
  • • Physics (projectile motion, gravity)
  • • Engineering (design curves, trajectories)
  • • Economics (profit maximization)
  • • Geometry (area optimization)
  • • Computer graphics (parabolic modeling)

Frequently Asked Questions

The quadratic formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), used to find the roots of any quadratic equation.

The discriminant (b² - 4ac) indicates root types:


  • Positive → 2 real roots
  • Zero → 1 real root
  • Negative → 2 complex roots

Use ( h = -b / 2a ) and ( k = f(h) ) to find the vertex (h, k).

The coefficient “a” controls the parabola’s direction and width.

  • a > 0 → opens upward
  • a < 0 → opens downward

The equation becomes linear, not quadratic.

Check the discriminant (b² - 4ac). If it’s ≥ 0, your equation has real roots.

  • Standard form: ax² + bx + c = 0
  • Vertex form: a(x - h)² + k

Yes, if the discriminant is negative, roots will include imaginary numbers (i).

It shows both the numeric roots and the vertex form, with an interactive parabola graph.

Absolutely. Enter decimal or fractional coefficients (e.g., 0.5, ¾), and the tool will compute precise results.