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Quadratic Formula Calculator

Solve second-degree quadratic equations ax² + bx + c = 0 for real and complex roots, discriminant, parabola vertex, focus, directrix, and step-by-step methods.

Algebra & Geometry ax² + bx + c = 0

Quadratic Formula Calculator

Solve second-degree quadratic equations for real and complex roots, discriminant ($b^2 - 4ac$), vertex coordinates, axis of symmetry, focus, directrix, and step-by-step methods.

Modify the values and click the Calculate button to use

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

• The solver computes exact roots, vertex coordinates, discriminant analysis, and step-by-step algebraic proofs.

1Quadratic Equation Solver

Enter coefficients for ax² + bx + c = 0

Coefficient (a)x² term

Must be non-zero for quadratic

Coefficient (b)x term

Linear coefficient

Constant (c)constant

Y-intercept constant

⚡ Quick Equation Presets:

Complete Guide to the Quadratic Formula, Parabola Geometry & Roots

An in-depth technical resource on algebraic derivations, discriminant classification, and geometric parabola properties.

1What is a Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x. The standard algebraic form is:

ax² + bx + c = 0, where a ≠ 0

Here, a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. When graphed on a Cartesian plane, every quadratic equation forms a symmetric U-shaped curve known as a parabola. The points where the parabola crosses the x-axis ($y = 0$) are the roots (also referred to as the solutions or x-intercepts).

2Derivation of the Quadratic Formula (Completing the Square)

The famous Quadratic Formula is derived systematically from ax² + bx + c = 0 by completing the square:

1. Subtract constant c: ax² + bx = -c

2. Divide by leading coefficient a: x² + (b/a)x = -c/a

3. Add (b / 2a)² = b² / (4a²) to both sides: x² + (b/a)x + b²/(4a²) = b²/(4a²) - c/a

4. Factor left side as a binomial square: (x + b/(2a))² = (b² - 4ac) / (4a²)

5. Take the square root of both sides: x + b/(2a) = ±√(b² - 4ac) / (2a)

6. Isolate x: x = [-b ± √(b² - 4ac)] / (2a)

3The Discriminant (Δ = b² - 4ac) & Nature of Roots

The expression underneath the square root, Δ = b² - 4ac, is called the discriminant. It completely determines the number and nature of roots without solving the entire equation:

Δ > 0 (Positive)

Two distinct real roots. The parabola crosses the x-axis at two different locations ($x_1 \ne x_2$).

Δ = 0 (Zero)

One repeated real root (multiplicity 2). The parabola touches the x-axis tangentially at its vertex ($x = -b/2a$).

Δ < 0 (Negative)

Two complex conjugate roots ($u \pm vi$). The parabola never touches or crosses the x-axis.

4Parabola Anatomy: Vertex, Focus & Directrix

Vertex & Axis of Symmetry

The vertex $(h, k)$ represents the turning point or peak/trough of the parabola:

• h = -b / (2a)

• k = c - b² / (4a) = f(h)

• Axis of Symmetry: vertical line x = h

Focus & Directrix

Geometrically, a parabola is the set of all points equidistant from the focus point and the directrix line:

• Focal Length: p = 1 / (4a)

• Focus Point: (h, k + 1/(4a))

• Directrix Line: y = k - 1/(4a)

5Real-World Applications of Quadratic Equations

Projectile Motion (Physics)

Objects launched under Earth's gravity follow parabolic trajectories described by $h(t) = -\frac12gt^2 + v_0 t + h_0$. Solving the quadratic finds the exact time of impact and maximum height.

Revenue & Profit Optimization

In economics, profit functions are commonly modeled as quadratic curves where revenue increases with price until demand drops. The vertex yields the optimal pricing strategy for maximum profit.

Optics & Satellite Dishes

Parabolic reflectors reflect incoming parallel light or radio waves directly into a single focal point (the focus), enabling satellite TV dishes, solar concentrators, and telescope mirrors.

Frequently Asked Questions

Can coefficient "a" equal zero in a quadratic equation?

No. If a = 0, the equation reduces to bx + c = 0, which is a first-degree linear equation rather than a quadratic. A true quadratic equation must have a non-zero leading coefficient (a ≠ 0).

What does a negative discriminant (Δ < 0) mean?

A negative discriminant means there are no real x-intercepts on the Cartesian plane. Instead, the equation has two complex conjugate roots involving the imaginary unit i (where i = √-1).

How do you find the vertex from standard form?

The x-coordinate of the vertex is h = -b / (2a). To find the y-coordinate k, substitute h into the original function: k = f(h) = a(h)² + b(h) + c = c - b²/(4a).

What is the difference between standard form and vertex form?

Standard form is written as y = ax² + bx + c, which clearly shows the y-intercept (0, c). Vertex form is written as y = a(x - h)² + k, which immediately displays the vertex coordinates (h, k) and axis of symmetry x = h.

Why does the ± sign exist in the quadratic formula?

Because every positive number has both a positive and negative square root (e.g., √9 = ±3 because 3² = 9 and (-3)² = 9). The ± symbol accounts for both possibilities.

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Understanding Quadratic Formula Calculator & Formula

Mathematical Formula: x = [-b ± √(b² - 4ac)] / (2a) | Vertex: (-b/2a, c - b²/4a) | Δ = b² - 4ac

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How to Use the Quadratic Formula Calculator

1 Enter coefficients a, b, and c for the standard form quadratic equation ax² + bx + c = 0.
2 Click Calculate to view real and complex conjugate roots, exact radical forms, and the interactive parabola graph.
3 Inspect comprehensive parabola properties including the Discriminant (Δ), Vertex (h, k), Axis of Symmetry, Focus, and Directrix.
4 Switch between step-by-step solving tabs: Quadratic Formula, Completing the Square, and Factoring methods.

FAQ & Help 3 questions