Quadratic Formula Solver

This tool solves any quadratic equation in the form ax² + bx + c = 0 by applying the quadratic formula, so you get the answer instantly instead of working through the algebra by hand. You enter the three coefficients, a, b and c, and it returns both roots, labelled real or complex depending on the discriminant, plus the discriminant itself and a plain English note on how many solutions the equation has. Alongside the roots you get full step-by-step working, from identifying the coefficients through to calculating the discriminant and applying the formula, and a set of extra properties: the sum and product of the roots for checking your answer against Vieta's formulas, the parabola's vertex coordinates, its axis of symmetry, which way the parabola opens, and the equation's factored form where one exists. A live equation preview redraws your equation as you type, and a summary verdict beneath the results explains in plain language what kind of roots you have got. Use it to check homework, verify factorising, find where a parabola crosses or touches the x-axis, or locate its turning point. Any real coefficients are accepted, including decimals and negative numbers, and results are shown to several decimal places, so very large or very small coefficients may show minor rounding, meaning you should substitute your roots back into the original equation to confirm the working.

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

1. Coefficients

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)
1
Two distinct real roots

Step-by-Step Working

Equation Properties

Standard formx2 − 5x + 6 = 0
Discriminant D1
Nature of rootsTwo distinct real roots
Sum of roots (x₁ + x₂)5
Product of roots (x₁ × x₂)6

Parabola Vertex

Vertex x-coordinate2.5
Vertex y-coordinate-0.25
Axis of symmetryx = 2.5
Parabola opensUpward (minimum at vertex)
Factored form(x − 3)(x − 2)
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 RootsExample
D > 0Two distinct real rootsx² - 5x + 6 = 0 gives x = 3 and x = 2
D = 0One repeated real rootx² - 4x + 4 = 0 gives x = 2 (repeated)
D < 0Two complex conjugate rootsx² + x + 1 = 0 gives x = -0.5 ± 0.866i

Worked Example

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

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

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.

Related Calculators

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.

Frequently asked questions

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).

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