Equation Solver with Steps
Solve quadratic equations (real or complex roots, discriminant, vertex, axis of symmetry) and 2-variable linear systems — with the working shown, not just the answer.
Interactive Client Prototype Sandbox
Disclaimer: This free tool is provided “as is,” without warranties of any kind, and is for general informational purposes only — not professional, legal, financial, medical, tax, or engineering advice. Results may contain errors; verify anything important independently and use at your own risk. We accept no liability for any loss or damage arising from its use. See our Terms of Use for details.
Step-by-Step Guide
Choose Quadratic or Linear System mode.
For a quadratic equation ax² + bx + c = 0, enter coefficients a, b, and c. The tool computes the discriminant (Δ = b² − 4ac), the roots (real or complex), the vertex coordinates, and the axis of symmetry, and shows each step of the derivation.
For a 2×2 linear system (two equations, two unknowns), enter the six coefficients in the form a1x + b1y = c1 and a2x + b2y = c2. The solver uses Cramer's Rule — computing determinants of the coefficient matrix and substitution matrices — to find x and y, with the working shown at each step. The tool also handles degenerate cases: if a = 0 the quadratic reduces to linear; if the system determinant is 0 the equations are either inconsistent (no solution) or dependent (infinitely many solutions).
Quadratic: x² − 5x + 6 = 0 (a=1, b=−5, c=6). Discriminant Δ = (−5)² − 4(1)(6) = 25 − 24 = 1. Since Δ > 0, two real roots: x = (5 ± √1) / 2. x₁ = 3, x₂ = 2. Vertex: x = −b/2a = 5/2 = 2.5; y = 6.25 − 12.5 + 6 = −0.25. Axis of symmetry: x = 2.5.
Linear system: 2x + 3y = 7 and x − y = 1. Determinant D = 2(−1) − 3(1) = −5. Dx = 7(−1) − 3(1) = −10 → x = 2. Dy = 2(1) − 7(1) = −5 → y = 1.
Who it's for
Students, teachers, engineers, and anyone checking algebra.
Core Features
- Quadratic solver: discriminant, real or complex roots, vertex, and axis of symmetry.
- 2×2 linear system solver via Cramer's rule.
- Step-by-step working shown for every result.
- Handles degenerate cases (a=0, no solution, infinitely many).
🛡️ No tracking — your inputs, keys, and details never leave this client sandbox.
What is the quadratic formula?
The quadratic formula gives the roots of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / (2a). The term b² − 4ac is called the discriminant (Δ). If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, the roots are complex conjugates (involving √(−Δ) = i√|Δ|).
What is a discriminant and what does it tell you?
The discriminant Δ = b² − 4ac determines the nature of the quadratic's roots without computing them. Δ > 0: two distinct real roots (the parabola crosses the x-axis twice). Δ = 0: one repeated real root (the parabola touches the x-axis at exactly one point — the vertex). Δ < 0: no real roots (the parabola does not cross the x-axis; roots are complex). The discriminant is the key to understanding the geometry of the parabola from its algebraic form.
What is Cramer's Rule?
Cramer's Rule is a method for solving a system of linear equations using determinants. For a 2×2 system a1x + b1y = c1 and a2x + b2y = c2, the solution is x = Dx/D and y = Dy/D, where D is the determinant of the coefficient matrix [[a1,b1],[a2,b2]], Dx is the determinant with the c column substituted for the x column, and Dy is the determinant with the c column substituted for the y column. If D = 0, the system is either inconsistent (no solution) or dependent (infinitely many).
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola y = ax² + bx + c. Its x-coordinate is x_v = −b/(2a), and the y-coordinate is found by substituting x_v into the equation: y_v = c − b²/(4a). If a > 0, the parabola opens upward and the vertex is the minimum; if a < 0, it opens downward and the vertex is the maximum. The vertex and the roots together fully characterize the parabola's geometry.
When the answer alone isn't enough
Picture this: a student submits a quadratic equation for a test and writes x = 3 and x = 2 as the answer. The grade comes back with zero marks. The answer is correct — but the working is missing, and the rubric requires the discriminant, the vertex calculation, and each step of the quadratic formula. This scenario plays out constantly: a homework checker, a test-prep tool, or a chat assistant that returns only the final answer is useless for learning. The value of an equation solver that shows its working is not the answer — it is the derivation that students can check their reasoning against and teachers can use as a marking reference.
The algebraic foundations
A quadratic equation is a degree-2 polynomial equation of the form ax² + bx + c = 0. The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) provides exact roots for any quadratic and has been known since al-Khwarizmi's 9th-century work 'Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala' (the word 'algebra' derives from 'al-jabr' in the title). The formula was expressed using geometric area arguments rather than algebraic notation at that time; modern symbolic notation came in the 16th and 17th centuries.
Complex roots and the imaginary unit
When the discriminant is negative, the square root in the quadratic formula produces √(negative number). This introduced the concept of imaginary numbers: the imaginary unit i is defined as √(−1), and complex numbers are written as a + bi. Complex roots always come in conjugate pairs (a + bi and a − bi) for polynomial equations with real coefficients. Although they have no 'physical' interpretation on the real number line, complex numbers are essential in electrical engineering, signal processing, quantum mechanics, and many other fields.
Determinants and Cramer's Rule
The determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc. It measures the signed area of the parallelogram formed by the matrix's column vectors. A determinant of zero means the matrix is singular — its column vectors are linearly dependent (parallel), so the matrix has no inverse. In the context of a linear system, a zero determinant means the two equations describe parallel lines (no intersection) or the same line (infinitely many intersections). Cramer's Rule expresses the solution as a ratio of determinants, making the structure of the solution transparent: each variable is the determinant of a modified matrix divided by the system determinant.