ƒ MATH · EIGHT CALCULATION MODES

Equation Solver

Solve a quadratic equation or two or three simultaneous linear equations. Enter coefficients to calculate real or complex quadratic roots. How to use it ↓

Solve a quadratic equation or two or three simultaneous linear equations. Enter coefficients to calculate real or complex quadratic roots.

Equation type

abc

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Enter an expression, then press Enter or =.

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History

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A little room for your working.
Your calculations will appear here.

About this result

Results round to 12 significant digits. Exact arithmetic fractions are available where supported.

Expression reference

sin(30) · asin(0.5)
ln(e) · log(8,2)
root(27,3) · 2^3
5! · nCr(5,2)
sinh(1) · conj(3+4i)
re(3+4i) · arg(3+4i)
1e-5 · 200+10%

Use exp(x) for eˣ and mod(a,b) for the remainder. Roots and logarithms may return complex values.

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How to use the equation solver

Choose a quadratic or a system of simultaneous linear equations.
For a quadratic, enter a, b and c, including zero coefficients.
For a linear system, enter one augmented matrix row per equation, with the constant term last.
Select Calculate and check the roots by substitution into the original equation or equations.

Formula and method

For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a)

Quadratic roots and the discriminant

The discriminant b²−4ac distinguishes two real roots, a repeated real root and a pair of complex roots. If a is zero, the expression is linear instead. When a and b are both zero, the equation is either an identity or a contradiction depending on c.

Simultaneous linear equations

Write each equation in coefficient order. For x+y=5 and 2x−y=1, enter rows 1 1 5 and 2 −1 1. A three-variable system uses three rows with coefficients for x, y, z and then the constant. The solver requires a unique solution and reports a residual for checking substitution.

ONE TOOL, DIFFERENT TASKS

Choose the mode for your calculation

Scientific Calculator

Powers, roots, logarithms, trigonometry, factorials, combinations and permutations. Decimal and exact arithmetic fraction displays, Ans and memory.

Complex Number Calculator

Calculate with i. Read the rectangular result, magnitude, argument and conjugate. Angle units apply to the argument and trigonometric functions.

Matrix Calculator

Add, subtract and multiply matrices up to 4 × 4. Find a determinant, inverse or transpose. Matrix B is needed only for binary operations.

Numerical Calculus Calculator

Approximate a definite integral or a derivative at a real point. This is numerical calculus; it does not produce symbolic antiderivatives.

Equation Solver

Solve a quadratic, including complex roots, or a system of two or three simultaneous linear equations. Degenerate quadratic inputs are handled as linear equations.

Statistics Calculator

Find count, sum, mean, median, range endpoints, variance and standard deviation for up to 1,000 values. Choose sample or population statistics.

Base-N Calculator

Convert signed integers of up to 128 bits between binary, octal, decimal and hexadecimal. Bitwise operations use unsigned 32-bit operands.

Function Table Calculator

Evaluate a real-valued function over a range of x values. Choose start, end and step; each table supports up to 201 rows.

TRY IT YOURSELF

Equation Solver worked examples

Solve a quadratic with two roots

x²−5x+6=0

x=2 or x=3

Factoring gives (x−2)(x−3)=0. Either factor can be zero.

Solve two simultaneous equations

x+y=5; 2x−y=1

x=2, y=3

Adding the equations gives 3x=6. Substituting x=2 into x+y=5 gives y=3.

Find conjugate complex roots

x²+1=0

x=−i or x=i

The square of either i or −i is −1, so both satisfy the equation.

READ THE RESULT WITH CONTEXT

Accuracy, notation and limits

This tool solves quadratics and systems of two or three linear equations. It does not rearrange arbitrary equations, solve general nonlinear systems or provide symbolic algebra steps.

Roots and linear solutions use numerical arithmetic. Large or ill-conditioned coefficients can lose precision, so substitute the results into the original equations. A small residual is useful evidence, not a guarantee for every input.

Read our calculation methodology ↗

COMMON QUESTIONS

Equation Solver FAQs

Can the solver return complex roots?

Yes. A negative quadratic discriminant produces conjugate complex roots, such as ±i for x²+1=0.

What if the quadratic coefficient a is zero?

The solver treats it as a linear equation bx+c=0. It also distinguishes an identity from an equation with no solution when b is zero.

Why does my linear system have no unique solution?

Dependent equations can give infinitely many solutions, while inconsistent equations give none. This solver needs a nonsingular coefficient system to return one unique solution.

Calculation library: math.js. This page is an independent CalcPebble tool.