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MATH · FREE ONLINE CALCULATOR

Algebraic Equation Solver Calculator

Solve linear, quadratic and cubic equations and systems of two or three linear equations. See exact fractions and square roots, complex roots, the discriminant and a graph of each solution.

Detailed optionsStep-by-step guideWorked examples
Linear, quadratic, cubic and simultaneous equations. Exact answers where possible.
01 / CHOOSE YOUR EQUATION

Enter the coefficients. Get every solution.

Results update instantly, with exact fractions and square roots alongside decimals.

++= 0
Your equation
x² − 5x + 6 = 0
Type a free-form equation instead

Type negative numbers with a minus sign: “x² − 5x + 6” means a = 1, b = −5, c = 6. A coefficient of 0 removes that term.

Two distinct real rootsx = 2 x = 3
Δ = 1 · two distinct real roots
Vertex: (2.5, −0.25)Factored: (x − 2)(x − 3)Sum of roots: 5 · product: 6
Where the graph crosses the x-axisMove across the graph to read values
024012345x = 2x = 3
02 / WHICH SOLVER DO I NEED?

Match your equation to the right method.

Pick a card to load that solver with an example.

Anything else? Brackets, fractions, roots, logs, trig or exponents: type it as you would write it.sqrt(x + 3) = x − 3
YOUR ANIMATED GUIDE

Follow the solution, step by step.

These steps use your current inputs. Choose a step or play the walkthrough.

STEP 1 / 4
x² − 5x + 6 = 0
1234

Identify a, b and c

Write the equation in the standard form ax² + bx + c = 0.

a = 1, b = −5, c = 6
Explore at your own pace

How to use this equation solver calculator

This algebraic equation solver handles the equation types used most in school and work: linear equations, quadratic equations, cubic equations and systems of two or three linear equations. Enter the coefficients and see every solution immediately, including repeated and complex roots, exact fractions and simplified square roots, the discriminant, the vertex of a parabola and a graph that marks each real solution.

Linear and quadratic equations

A linear equation such as 3x − 4 = 11 has x only to the first power and is solved by isolating x: x = (c − b) ÷ a. A quadratic equation ax² + bx + c = 0 has x squared. Its discriminant Δ = b² − 4ac decides the type of solution before you solve: positive gives two real roots, zero gives one repeated root, and negative gives two complex conjugate roots. The quadratic formula x = (−b ± √Δ) ÷ 2a then gives the roots.

Cubic equations

A cubic equation ax³ + bx² + cx + d = 0 always has three roots when repeated and complex roots are counted, and at least one of them is real. With whole-number coefficients the calculator first applies the rational root test, which tries ± factors of d divided by factors of a. Each root found is divided out exactly, so remaining roots can often be written with square roots. Otherwise the roots are found numerically and polished with Newton’s method.

Systems of linear equations

A system asks for values that make every equation true at the same time. Graphically, a 2 × 2 system is two straight lines and the solution is where they cross. If the determinant of the coefficients is not zero, Cramer’s rule gives one unique solution. A determinant of zero means the lines are parallel (no solution) or the same line (infinitely many solutions); elimination tells the two cases apart.

Equation Solver formulas, explained

Linear equation

ax + b = c ⇒ x = (c − b) ÷ a

If a = 0, the equation is either always true or never true.

Quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Δ = b² − 4ac. Vertex at x = −b ÷ 2a. Sum of roots −b/a, product c/a.

Cubic discriminant

Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²

Δ > 0: three distinct real roots. Δ = 0: a repeated root. Δ < 0: one real and two complex roots.

Cramer’s rule (2 × 2)

x = (c₁b₂ − c₂b₁) ÷ D, y = (a₁c₂ − a₂c₁) ÷ D

D = a₁b₂ − a₂b₁ must not be zero.

PUT THE FORMULA TO WORK

Worked examples you can try.

Load an example into the calculator, then change one value to explore.

EXAMPLE 01

A two-root quadratic

x² − 5x + 6 = 0 has Δ = 1, so two real roots: x = 2 and x = 3. Factored form: (x − 2)(x − 3).

EXAMPLE 02

Complex roots

x² + 2x + 5 = 0 has Δ = −16. The roots are −1 − 2i and −1 + 2i, so the parabola never touches the x-axis.

EXAMPLE 03

A cubic with three rational roots

x³ − 7x + 6 = 0 factors as (x − 1)(x − 2)(x + 3), giving x = −3, 1 and 2.

EXAMPLE 04

Three equations, three unknowns

2x + 3y − z = 1, x − y + 2z = 8 and 3x + y + z = 9 meet at x = 2, y = 0, z = 3.

MAKE THE NUMBERS MAKE SENSE

Common mistakes to avoid

  • Forgetting the sign of a coefficient: x² − 5x + 6 has b = −5, not 5.
  • Not moving every term to one side before reading a, b and c.
  • Dividing both sides by x, which can lose the solution x = 0.
A LITTLE MORE CLARITY

Frequently asked questions

Can this solver find complex roots?

Yes. Quadratic and cubic equations with a negative discriminant show their complex conjugate roots in the form a ± bi.

Which equation types are supported?

Linear (ax + b = c), quadratic (ax² + bx + c = 0), cubic (ax³ + bx² + cx + d = 0) and systems of two or three linear equations.

What does a negative discriminant mean?

There is no real number whose square is negative, so the quadratic has no real roots and the graph does not cross the x-axis. The calculator shows the two complex roots, which are conjugates of each other.

Why does the calculator show a square root instead of a decimal?

When the coefficients are whole numbers or simple fractions, roots such as (3 + √5)/2 are exact. The decimal approximation is shown next to it so you can check the size of the answer.

What happens if a is zero?

The equation drops a degree. A quadratic with a = 0 is linear, and a linear equation with a = 0 is either true for every x or for none. The calculator tells you which case applies.

Can I solve equations that are not in standard form?

Yes. Use the X Pro Super Solver to type an equation such as 3(x − 2) + 4 = 2x² − 5 exactly as it is written. It expands and rearranges the equation before solving.

How accurate are numerical roots?

Numerical roots are polished with Newton’s method and are typically accurate to more than ten significant digits. Displayed values are rounded to six decimal places.

Methodology & assumptions

Calculations run in your browser using the formulas and assumptions explained on this page. Intermediate values retain full floating-point precision; displayed amounts are rounded. Undefined operations such as division by zero are rejected, every candidate solution is substituted back into the original equation, and numerical answers are refined to about 15 significant digits. Sharing an input link places those inputs in the URL.

Further reading:Quadratic formulaCramer’s rule
KEEP EXPLORING

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