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Solve linear, quadratic, and 2×2 linear systems with a numerical residual check on every result.
What this tool helps you answer
Solves linear equations (ax+b=c), quadratic equations (ax²+bx+c=0), and 2×2 linear systems using direct algebra and Cramer's rule, then verifies each answer with a residual check.
Use the model, assumptions, metrics, and warnings together before acting on the output.
Next step
Solve linear, quadratic, and 2×2 linear systems with a numerical residual check on every result.
Editorial review
This page combines the live tool, input guidance, worked examples, and operating limits so Equation Solver stays useful even before users interact with the calculator.
Reviewed by Klartext Tools against the current Equation Solver workflow on 2026-03-06.
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x² − 5x + 6 = 0 has roots at x = 2 and x = 3. This is a clean example to confirm residual behaviour.
The solver returns x₁ = 3, x₂ = 2, discriminant = 1, and a residual near zero for both roots.
After running this example, try changing c to 7 to see what happens when the discriminant turns negative.
2x + y = 5 and x − y = 1 is a classic simultaneous equation pair with a unique solution.
The solver returns x = 2, y = 1 with residuals near zero for both equations.
Change a2 to 2 and b2 to 0.5 to create a singular system and observe the determinant-zero message.
Select the equation type first, then enter the coefficients that match your problem.
Select Linear for ax + b = c, Quadratic for ax² + bx + c = 0, or 2×2 System for two simultaneous linear equations.
Fill in the numeric values for each coefficient. For the quadratic mode the default shows x² − 3x + 2 = 0, which has two real roots at x = 1 and x = 2.
The result panel shows the solution alongside a residual value. A residual near zero confirms the answer is numerically correct.
If the discriminant is negative the solver returns complex roots. If the 2×2 system determinant is zero the solver reports whether the system has no solution or infinite solutions.
Load a simple quadratic to see how roots, discriminant, and residual appear together in the result panel.
x² − 5x + 6 = 0 has roots at x = 2 and x = 3. This is a clean example to confirm residual behaviour.
Sample inputs
Sample outcome: The solver returns x₁ = 3, x₂ = 2, discriminant = 1, and a residual near zero for both roots.
After running this example, try changing c to 7 to see what happens when the discriminant turns negative.
2x + y = 5 and x − y = 1 is a classic simultaneous equation pair with a unique solution.
Sample inputs
Sample outcome: The solver returns x = 2, y = 1 with residuals near zero for both equations.
Change a2 to 2 and b2 to 0.5 to create a singular system and observe the determinant-zero message.
Solving equations by hand is error-prone for anything beyond a simple linear case, and most symbolic solvers return a result without the context needed to trust it. This solver shows a residual check alongside every result, the numerical verification that the answer actually satisfies the original equation, so you are not just reading back a number the formula produced. It is useful for double-checking your own work, validating model assumptions before writing code, or building intuition for systems of equations without relying on a full computer algebra system.
The equation solver handles three classes of algebraic problem in a single interface. Linear mode solves ax + b = c for x using direct rearrangement. Quadratic mode applies the quadratic formula to ax² + bx + c = 0 and returns both roots, real or complex, along with the discriminant value. System mode uses Cramer's rule to solve two simultaneous linear equations in two unknowns. Every solution includes a residual check: the computed answer is substituted back into the original equation and the difference between the two sides is reported, giving you numerical confidence in the result.
Linear equations have exactly one solution unless the coefficient of x is zero, in which case the equation is either always true (infinite solutions) or never true (no solution). Quadratic equations are governed by the discriminant D = b² − 4ac: when D > 0 there are two distinct real roots, when D = 0 there is one repeated root, and when D < 0 the roots are complex conjugates. For 2×2 linear systems, Cramer's rule expresses each unknown as a ratio of determinants. When the system determinant is zero the equations are either parallel (inconsistent) or identical (dependent) and no unique solution exists.
This guide explains how to solve and interpret quadratic equations from setup to root analysis.
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