Equation Solver
Solve linear, quadratic and systems of equations step by step, free online.
Results update as you type.
About this calculator
An equation solver finds the values that satisfy an equation. This tool solves linear equations, quadratic equations (via the quadratic formula, including complex roots) and systems of two linear equations. Choose the type, enter the coefficients, and get the solution with the key working shown.
For a linear equation ax + b = 0 it simply rearranges to x = -b/a. For a quadratic ax² + bx + c = 0 it applies the quadratic formula, first computing the discriminant b² − 4ac: a positive discriminant gives two real roots, zero gives one repeated root, and a negative discriminant gives a pair of complex-conjugate roots which it reports rather than declaring "no solution". For a 2×2 system it solves the two lines simultaneously to find the single point where they cross.
For a worked example, solving x² − 5x + 6 = 0 gives a discriminant of 25 − 24 = 1, so the roots are x = (5 ± 1)/2, that is x = 3 and x = 2. By contrast x² + 1 = 0 has discriminant -4 and returns the complex roots x = i and x = -i. Students use it to check algebra homework, find where a parabola crosses the x-axis, or solve the pair of equations behind a word problem.
Frequently asked questions
What types of equations can it solve?
Linear equations (ax + b = 0), quadratics (ax² + bx + c = 0, including complex roots), and 2×2 systems of linear equations.
Does it show complex roots?
Yes. When a quadratic has a negative discriminant, the solver reports the two complex conjugate roots rather than saying there is no solution.
How do I solve a quadratic like x² − 5x + 6 = 0?
Choose the quadratic type and enter a = 1, b = -5, c = 6. The solver applies the quadratic formula and returns x = 2 and x = 3, the two values that make the expression zero.
What is the discriminant and why does it matter?
The discriminant is b² − 4ac. If it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the roots are a complex conjugate pair. The solver uses it to decide which case applies.
How do I solve a system of two equations?
Select the 2×2 system type and enter the coefficients of both equations. The solver finds the single (x, y) point where the two lines meet, or tells you when they are parallel and never cross.
How is this different from the graphing calculator?
The equation solver returns the exact solution values algebraically. The graphing calculator instead draws the curves so you can see the roots as x-axis crossings or the intersection of two graphs.
Results are estimates for general guidance only, not financial, medical or tax advice.