EQUATION SOLVER GUIDE

How to solve linear, quadratic, cubic, and system equations

Equation solving starts by identifying the equation form. A one-variable equation may be linear, quadratic, or cubic, while a simultaneous system is usually written as a coefficient matrix A·x = b.

FORMULA

Linear: ax + b = 0; Quadratic: ax² + bx + c = 0; Cubic: ax³ + bx² + cx + d = 0; System: A·x = b

WORKED EXAMPLE

For x² − 5x + 6 = 0, the quadratic roots are x = 2 and x = 3. For a system, write each row as coefficients followed by the right-side value, then solve the matrix system.

STEP BY STEP

Identify the equation type first

If the highest power is x, use the linear workflow. If the highest power is x², use the quadratic workflow. If the highest power is x³, use the cubic workflow. If there are several equations with several unknowns, use the linear system workflow.

Enter coefficients in the correct order

For one-variable equations, enter a, b, c, and d exactly as they appear in standard form. For systems, keep variables in the same order on every row and put constants on the right side of the augmented matrix.

Use the discriminant for quadratics

The quadratic discriminant Δ = b² − 4ac explains the result type. Positive means two real roots, zero means one repeated real root, and negative means complex roots.

Use system methods for simultaneous equations

Gaussian elimination, Gauss-Jordan RREF, matrix inverse, and Cramer’s rule are different ways to solve A·x = b. Row reduction is usually the safest general method, while Cramer’s rule is mainly for small systems.

Classify special system results

A system can have one solution, no solution, or infinitely many solutions. RREF and rank checks help show whether rows contradict each other or whether one variable is free.

Check whether another SolveGrid tool is better

Use Polynomial Solver for degree 4 to 10 roots, Graphing Calculator for visual intercepts and tables, and Matrix Calculator when you want deeper REF, RREF, determinant, rank, or inverse inspection.