📊Quadratic Equation Solver

This quadratic formula calculator lets you solve quadratic equations online in seconds. Enter the coefficients a, b, and c for any equation in the form ax squared plus bx plus c equals zero, and the calculator returns both roots, the discriminant, vertex, axis of symmetry, and parabola direction. It handles real roots, repeated roots, and complex roots automatically.

Quadratic Formula Calculator with Step-by-Step Solution

The quadratic formula is the universal method for solving any quadratic equation. It states that the two solutions for x are:

x = (-b plus or minus the square root of (b squared minus 4ac)) divided by (2a)

The plus-or-minus sign means the formula produces two answers. One uses addition inside the numerator and the other uses subtraction, giving the two roots of the equation. To use this calculator, identify the three coefficients from your equation and enter them into the fields labeled a, b, and c.

For the equation 2x squared minus 4x minus 6 equals zero, the coefficients are a = 2, b = -4, and c = -6. The calculator solves for both roots, displays the discriminant, and identifies the vertex of the corresponding parabola.

How to Solve ax Squared Plus bx Plus c Equals Zero

There are several algebraic methods for solving quadratic equations, each with advantages depending on the problem.

Factoring

Factoring works when the quadratic can be rewritten as a product of two linear expressions. For x squared minus 5x plus 6 equals zero, the factored form is (x minus 2)(x minus 3) equals zero, giving roots x = 2 and x = 3. Factoring is the fastest approach when the roots are whole numbers, but many quadratics do not factor neatly over integers.

Completing the Square

This method rewrites the equation in the form a(x plus h) squared plus k equals zero. It is the technique used to derive the quadratic formula and is especially useful when identifying the vertex of the parabola. When the leading coefficient a equals 1, completing the square is straightforward to apply by hand.

The Quadratic Formula

The formula works for every quadratic equation without exception, including those with irrational or complex roots. It is the most reliable method when factoring is not obvious and is the approach this calculator uses internally.

Discriminant Calculator for Quadratic Equations

The discriminant is the expression b squared minus 4ac that appears under the square root in the quadratic formula. Its value tells you the nature of the roots before you solve the full equation.

• Positive discriminant: Two distinct real roots. The parabola crosses the x-axis at two different points.
• Zero discriminant: One repeated real root, also called a double root. The parabola just touches the x-axis at a single point without crossing it.
• Negative discriminant: No real roots. The solutions are complex numbers. The parabola sits entirely above or below the x-axis without intersecting it.

Checking the discriminant first is a useful shortcut. If you need real solutions and the discriminant is negative, there are none and no further algebra is necessary.

The Parabola: Vertex, Axis of Symmetry, and Direction

Every quadratic equation corresponds to a parabola when graphed. The vertex is the highest or lowest point of the parabola and has x-coordinate equal to -b divided by (2a). Substituting this back into the equation gives the y-coordinate of the vertex.

The axis of symmetry is the vertical line passing through the vertex, dividing the parabola into two mirror halves. When the coefficient a is positive, the parabola opens upward and the vertex is its minimum point. When a is negative, the parabola opens downward and the vertex is its maximum point.

The y-intercept, where the parabola crosses the vertical axis, is always the point (0, c), which is simply the constant term of the equation.

Real-World Applications of Quadratic Equations

Quadratic equations model many real situations across science, engineering, and business.

• Projectile motion: The height of an object thrown upward follows a quadratic path over time. Setting height equal to zero and solving gives the time when the object lands.
• Revenue optimization: When price and demand are linearly related, total revenue becomes quadratic. The vertex of the resulting parabola gives the price that maximizes revenue.
• Geometry: Problems involving the area of rectangles with a fixed perimeter lead naturally to quadratic equations.
• Engineering: Structural load analysis and circuit design both use quadratic relationships to find critical operating points.