What is a quadratic equation and standard form?

A quadratic equation is a second-order polynomial equation in a single variable x, with standard algebraic form ax² + bx + c = 0, where a, b, and c are constant real coefficients and a ≠ 0. The graph of every real quadratic function is a U-shaped symmetric curve known as a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

What is the Discriminant (Δ = b² - 4ac) and what does it tell us?

The discriminant Δ (Delta) is the expression underneath the square root in the quadratic formula: Δ = b² - 4ac. It determines the nature and number of roots: (1) If Δ > 0, the equation has two distinct real roots and the parabola intersects the x-axis at two points; (2) If Δ = 0, the equation has exactly one real repeated root and the parabola's vertex touches the x-axis; (3) If Δ < 0, the equation has two complex conjugate roots (with imaginary unit i) and the parabola does not touch the x-axis.

What happens if the leading coefficient a is zero (a = 0)?

If a = 0, the quadratic term ax² vanishes, reducing the equation to a first-degree linear equation: bx + c = 0. Its single solution is x = -c / b (provided b ≠ 0). If both a = 0 and b = 0, the equation either has no solution (if c ≠ 0, an algebraic contradiction) or infinitely many solutions (if c = 0, an identity).

How do you find the vertex (h, k) of a parabola?

The vertex of a parabola represents its highest point (maximum, when a 0). The x-coordinate of the vertex h lies on the vertical axis of symmetry at h = -b / (2a). The y-coordinate k is obtained by evaluating the function at h: k = c - b² / (4a), or k = (4ac - b²) / (4a) = -Δ / (4a).

How are complex or imaginary roots represented?

When the discriminant is negative (Δ < 0), the square root of Δ involves the imaginary unit i (where i = √(-1)). The roots take the complex conjugate form x = (-b / 2a) ± (√(|Δ|) / 2a) i. For example, x² - 4x + 13 = 0 yields Δ = -36, giving complex roots x = 2 ± 3i.

Are my calculations and polynomial solutions private?

Yes. 100% of arithmetic, discriminant computations, step-by-step derivations, and SVG plotting execute directly in your web browser. No equations, coefficients, or schoolwork data are transmitted to an external server.

Quadratic Equation Solver & Parabola Grapher

Q: What is the Discriminant (Δ = b² - 4ac) and what does it tell us?

The discriminant Δ (Delta) is the expression underneath the square root in the quadratic formula: Δ = b² - 4ac. It determines the nature and number of roots: (1) If Δ > 0, the equation has two distinct real roots and the parabola intersects the x-axis at two points; (2) If Δ = 0, the equation has exactly one real repeated root and the parabola's vertex touches the x-axis; (3) If Δ < 0, the equation has two complex conjugate roots (with imaginary unit i) and the parabola does not touch the x-axis.

Q: What happens if the leading coefficient a is zero (a = 0)?

If a = 0, the quadratic term ax² vanishes, reducing the equation to a first-degree linear equation: bx + c = 0. Its single solution is x = -c / b (provided b ≠ 0). If both a = 0 and b = 0, the equation either has no solution (if c ≠ 0, an algebraic contradiction) or infinitely many solutions (if c = 0, an identity).

Q: How do you find the vertex (h, k) of a parabola?

The vertex of a parabola represents its highest point (maximum, when a 0). The x-coordinate of the vertex h lies on the vertical axis of symmetry at h = -b / (2a). The y-coordinate k is obtained by evaluating the function at h: k = c - b² / (4a), or k = (4ac - b²) / (4a) = -Δ / (4a).

Q: How are complex or imaginary roots represented?

When the discriminant is negative (Δ < 0), the square root of Δ involves the imaginary unit i (where i = √(-1)). The roots take the complex conjugate form x = (-b / 2a) ± (√(|Δ|) / 2a) i. For example, x² - 4x + 13 = 0 yields Δ = -36, giving complex roots x = 2 ± 3i.

Q: Are my calculations and polynomial solutions private?

Yes. 100% of arithmetic, discriminant computations, step-by-step derivations, and SVG plotting execute directly in your web browser. No equations, coefficients, or schoolwork data are transmitted to an external server.

Solve standard quadratic equations ax² + bx + c = 0 with real-time discriminant analysis, step-by-step formula derivations, vertex computation, and interactive parabola plotting.

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Equation
Roots
Vertex
SVG Plot

Standard Quadratic Form

ax² + bx + c = 0

Explore Practice Equations

Coefficient a (x²)Quadratic term coefficient (a ≠ 0 for parabola)

Coefficient b (x)Linear term coefficient

Constant c (y-intercept)Constant numerical value

Calculated Roots (Solutions)

x₁ =3

x₂ =2

Two Distinct Real Roots (Δ > 0)Opens up (a > 0)

Discriminant (Δ)1

Vertex (h, k)(2.5, -0.25)

Interactive Parabola Plot (y = 1x² - 5x + 6)

Vertex Roots

Formula & Step-by-Step Algebraic Derivation

Step 1: Identify constant coefficients:

a = 1, b = -5, c = 6

Step 2: Calculate the Discriminant (Δ):

Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - (24) = 1

Step 3: Substitute coefficients into the Quadratic Formula:

x = [ -(-5) ± √(1) ] / [ 2(1) ]

Step 4: Solve for roots:

x₁ = [ 5 + 1 ] / 2 = 3, x₂ = [ 5 - 1 ] / 2 = 2

Step 5: Calculate Parabola Vertex & Line of Symmetry:

h = -b / (2a) = -(-5) / (2 × 1) = 2.5, k = f(h) = -0.25

What is a Quadratic Equation?

A quadratic equation is a second-order polynomial equation involving a single variable x raised to the power of two. In standard algebraic form, it is written as:

ax² + bx + c = 0

where a, b, and c are real numbers, and a ≠ 0.

The graph of every real quadratic function is a symmetrical geometric curve known as a parabola. If the leading coefficient a > 0, the parabola opens upward with its vertex forming a minimum; if a < 0, it opens downward with its vertex forming a maximum.

How to Use the Quadratic Formula Calculator?

Enter Coefficients

Input your quadratic coefficients a, b, and constant c. You can also click any practice preset to load sample textbook problems.

Evaluate Roots & Vertex

The solver computes the discriminant (Δ), displays exact roots x₁ and x₂, computes vertex coordinates (h, k), and highlights the axis of symmetry.

Inspect Plot & Steps

Review the step-by-step substitution into the quadratic formula and explore the responsive SVG parabola plot with marked roots and vertex.

Understanding the Discriminant (Δ = b² - 4ac)

The quantity under the square root radical sign in the quadratic formula is called the Discriminant (denoted by Greek letter Delta, Δ). The sign of Δ dictates the number and type of roots:

Case 1: Δ > 0

Two Distinct Real Roots

The parabola crosses the x-axis at two distinct real x-intercepts. Both roots can be evaluated directly with arithmetic.

Case 2: Δ = 0

One Repeated Real Root

The parabola is tangent to the x-axis. Its vertex lies exactly on the x-axis, creating a single double root at x = -b / (2a).

Case 3: Δ < 0

Two Complex Conjugate Roots

The parabola never touches or crosses the x-axis. Roots are complex numbers in the form of Re ± Im·i involving the imaginary unit i.

Frequently Asked Questions

A quadratic equation is a second-degree polynomial equation written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of every quadratic equation is a symmetrical U-shaped curve called a parabola.

100% Client-Side Privacy: All algebraic computations, discriminant analyses, and SVG parabola renderings occur exclusively in your local browser. No mathematical inputs or equations are ever transmitted to any remote server.

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Quadratic Equation Solver & Parabola Grapher

Interactive Parabola Plot (y = 1x² - 5x + 6)

Formula & Step-by-Step Algebraic Derivation

What is a Quadratic Equation?

How to Use the Quadratic Formula Calculator?

Enter Coefficients

Evaluate Roots & Vertex

Inspect Plot & Steps

Understanding the Discriminant (Δ = b² - 4ac)

Two Distinct Real Roots

One Repeated Real Root

Two Complex Conjugate Roots

Frequently Asked Questions

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Formula & Step-by-Step Algebraic Derivation

What is a Quadratic Equation?

How to Use the Quadratic Formula Calculator?

Enter Coefficients

Evaluate Roots & Vertex

Inspect Plot & Steps

Understanding the Discriminant (Δ = b² - 4ac)

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Frequently Asked Questions