What is the quadratic formula?

The quadratic formula gives the roots of any quadratic equation in the form ax² + bx + c = 0. The formula is x = (-b ± √(b² - 4ac)) / (2a). The expression under the square root, b² - 4ac, is called the discriminant. If the discriminant is positive the equation has two distinct real roots. If it equals zero there is exactly one repeated real root. If it is negative the equation has two complex (imaginary) roots.

What does the discriminant tell you?

The discriminant D = b² - 4ac reveals the nature of a quadratic equation's roots without fully solving it. A positive discriminant means two distinct real roots (the parabola crosses the x-axis at two points). A discriminant of zero means one repeated real root (the parabola just touches the x-axis at its vertex). A negative discriminant means no real roots; instead there are two complex conjugate roots of the form p + qi and p - qi.

How do I factorise a quadratic once I know the roots?

If the roots of ax² + bx + c = 0 are x₁ and x₂, the factored form is a(x - x₁)(x - x₂). For example, x² - 5x + 6 = 0 has roots x = 3 and x = 2, so it factors as (x - 3)(x - 2). You can verify this by expanding: x² - 2x - 3x + 6 = x² - 5x + 6. When the leading coefficient a is not 1, include it outside the brackets: 2x² - 7x + 3 = 2(x - 3)(x - 0.5) or equivalently (2x - 1)(x - 3).

Quadratic Formula Solver

Enter the coefficients a, b, and c of your quadratic equation in the form ax² + bx + c = 0. The solver applies the quadratic formula to find both roots, shows the discriminant, and gives full step-by-step working. Both real and complex roots are supported.

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Standard formula Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

x = (-b ± √(b² - 4ac)) / (2a)

1. Coefficients

a (coefficient of x², cannot be 0)

b (coefficient of x)

c (constant term)

Please enter valid numbers. The value of a cannot be zero.

2. Equation Preview

Your equation displayed as entered:

x² − 5x + 6 = 0

Enter decimal values such as 1.5 or negative values such as −3. The solver handles any real coefficients.

Roots of the Equation

Root 1 (x₁)
3
Real root

Root 2 (x₂)
2
Real root

Discriminant (b² − 4ac)

Two distinct real roots

Step-by-Step Working

Equation Properties

Standard form-

Discriminant D-

Nature of roots-

Sum of roots (x₁ + x₂)-

Product of roots (x₁ × x₂)-

Parabola Vertex

Vertex x-coordinate-

Vertex y-coordinate-

Axis of symmetry-

Parabola opens-

Factored form-

Result: Enter your coefficients above.

The Quadratic Formula

A quadratic equation is any equation of the form ax² + bx + c = 0, where a is not zero. The quadratic formula gives the roots (solutions) of this equation directly from its coefficients:

x = (-b ± √(b² - 4ac)) / (2a)

The formula is derived by completing the square on the general form. It always produces the correct roots regardless of whether they are integers, fractions, irrational numbers, or complex numbers.

The Discriminant

The expression D = b² - 4ac inside the square root is called the discriminant. It determines the nature of the roots without you having to compute them:

Discriminant (D)	Nature of Roots	Example
D > 0	Two distinct real roots	x² - 5x + 6 = 0 gives x = 3 and x = 2
D = 0	One repeated real root	x² - 4x + 4 = 0 gives x = 2 (repeated)
D < 0	Two complex conjugate roots	x² + x + 1 = 0 gives x = -0.5 ± 0.866i

Worked Example

Solve x² − 5x + 6 = 0 (where a = 1, b = −5, c = 6).

Identify: a = 1, b = −5, c = 6
Discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1
Square root: √1 = 1
Root 1: x₁ = (−(−5) + 1) / (2 × 1) = (5 + 1) / 2 = 6 / 2 = 3
Root 2: x₂ = (−(−5) − 1) / (2 × 1) = (5 − 1) / 2 = 4 / 2 = 2
Check: (3)² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓

Sum and Product of Roots

By Vieta's formulas, the sum and product of the roots of ax² + bx + c = 0 can be found directly from the coefficients without solving the equation:

Sum of roots: x₁ + x₂ = −b / a
Product of roots: x₁ × x₂ = c / a

For x² − 5x + 6 = 0: sum = −(−5)/1 = 5; product = 6/1 = 6. Verify: 3 + 2 = 5 and 3 × 2 = 6.

Vertex of the Parabola

The graph of y = ax² + bx + c is a parabola. Its vertex (turning point) sits at x = −b / (2a) and y = c − b²/(4a). For a positive value of a the parabola opens upward and the vertex is a minimum. For a negative value of a it opens downward and the vertex is a maximum. The axis of symmetry is the vertical line x = −b / (2a), which sits exactly halfway between the two roots when they exist.

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Method: The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) is derived by completing the square on ax² + bx + c = 0. Complex roots are expressed in the form p ± qi where i = √(-1). Vertex coordinates use x = -b/(2a) and y = c - b²/(4a).

This solver handles any real coefficients and returns exact decimal results. For equations with very large or very small coefficients, rounding may occur in the displayed values. Results should be verified by substituting back into the original equation.

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