Quadratic Formula Calculator
Solve any quadratic equation with the quadratic formula, with exact roots, the discriminant, factored form and vertex, solve linear equations and 2 × 2 systems, and simplify arithmetic expressions exactly.
Quadratic formula calculator: ax² + bx + c = 0
Equation solver: linear equations and 2 × 2 systems
Solve ax + b = c:
Solve a₁x + b₁y = c₁ and a₂x + b₂y = c₂:
Simplification calculator (exact fractions, BODMAS)
The quadratic formula
For ax² + bx + c = 0 (a ≠ 0): x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac tells you what kind of roots to expect: positive, two real roots; zero, one repeated root; negative, two complex roots.
- x² − 5x + 6 = 0: D = 25 − 24 = 1, so x = (5 ± 1) ÷ 2 = 3 or 2, and it factors as (x − 2)(x − 3).
- x² − 2x − 1 = 0: D = 8 = 4 × 2, so x = (2 ± 2√2) ÷ 2 = 1 ± √2 (2.4142 and −0.4142).
- x² + 4x + 5 = 0: D = −4, so x = −2 ± i.
The vertex of the parabola y = ax² + bx + c is at x = −b ÷ 2a; for x² − 5x + 6 it's (2.5, −0.25). This quadratic equation solver keeps answers exact (fractions and simplified square roots) whenever a, b and c are whole numbers or fractions.
Equation solver
A linear equation ax + b = c has the single solution x = (c − b) ÷ a: 3x + 7 = 22 gives x = 5. A system of two linear equations is solved here with Cramer's rule: 2x + 3y = 12 and x − y = 1 give x = 3, y = 2. For any other equation in x (cubic, trig, exponential), use the algebra solver on the scientific calculator.
Simplification calculator: order of operations
Simplifying an arithmetic expression follows BODMAS (PEMDAS in the US): brackets, orders (powers), division and multiplication left to right, then addition and subtraction left to right; "of" means multiply. 3/4 + 5/6 × 2 = 3/4 + 5/3 = 9/12 + 20/12 = 29/12 = 2 5/12. This simplifying calculator works in exact fractions, so there are no rounding errors.
Common questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a for ax² + bx + c = 0.
What are the roots of x² − 5x + 6?
x = 2 and x = 3; it factors as (x − 2)(x − 3).
What does the discriminant tell you?
b² − 4ac > 0: two real roots; = 0: one repeated root; < 0: two complex roots.
How do I solve 2x + 3y = 12 and x − y = 1?
x = 3, y = 2.
What is 3/4 + 5/6 × 2?
29/12, or 2 5/12, because multiplication comes before addition.
Sources
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General information, not financial, tax or legal advice. Check important numbers with the official source before acting on them.