yCalculator

Quadratic Formula Calculator

Last updated: April 2026

Coefficients

x^2 - 5x + 6 = 0

Δ = -5² - 4 × 1 × 6 = 1

Two distinct real solutions

Solutions

x1

3

x2

2

Factored form

(x - 3)(x - 2) = 0

Sum of roots

5

Product of roots

6

Step-by-step working
  1. Identify coefficients: a = 1, b = -5, c = 6
  2. Delta = b^2 - 4ac
  3. Delta = (-5)^2 - 4(1)(6)
  4. Delta = 25 - 24 = 1
  5. x = (-b +/- sqrt(Delta)) / 2a
  6. x = (5 +/- 1) / 2
  7. x1 = 3, x2 = 2
  8. Verification x=3: 1(9) + -5(3) + 6 = 0
  9. Verification x=2: 1(4) + -5(2) + 6 = 0

The quadratic formula explained

The quadratic formula solves any equation in the form ax² + bx + c = 0, even when it cannot be factored neatly.

What is the discriminant?

The discriminant, b² - 4ac, tells you how many roots the quadratic has and whether they are real or complex.

How to factor quadratic equations

If the roots are simple integers, the equation can often be written as a product of two brackets.

Vertex form and parabolas

Vertex form shows the turning point of the parabola directly. If a is positive it opens upward; if a is negative it opens downward.

What does this mean?

This calculator is designed to help you understand the likely number before you make a decision or start an application.

Your result should be checked against official UK guidance, especially if your circumstances include dependants, exemptions, prior leave, or a complex immigration history.

Treat the figure as a planning tool rather than legal advice. Where the answer affects an application deadline or major payment, speak to an authorised adviser.

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