Equation Solver
Solve linear equations, quadratic equations, and systems of two equations instantly. Enter coefficients and get exact solutions with full step-by-step working.
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★ Key Features
3 Equation Types
Linear, quadratic, and simultaneous systems all in one tool.
Full Step-by-Step
Formula substitution and verification shown for every solution.
Instant Solutions
Results update the moment you enter or change any coefficient.
Mobile Optimized
Large touch-friendly coefficient inputs — easy to use on phones.
✎ Real-World Example
Δ = b²−4ac = 400 − (−20) = 420
t = (−20 ± √420) / (−10)
t₁ ≈ 4.049, t₂ ≈ −0.049 (negative rejected)
Q Frequently Asked Questions
Equation Solver - Solve Linear, Quadratic, and System Equations with Full Steps
"Solve for x" is probably the single most repeated instruction in algebra class, but it means genuinely different things depending on what kind of equation is sitting in front of you. A straightforward equation like 3x + 5 = 20 solves in two easy steps. Add a squared term, and suddenly you're pulling out the quadratic formula. Add a second equation with a second variable, and now you're solving a system rather than a single equation. An equation solver that treats these as three distinct problem types — rather than one generic "solve for x" button — genuinely reflects how differently these problems actually work.
The real value in a tool like this isn't just speed, either. It's confirming you're using the right method for the right equation type, and seeing the full working laid out so a mistake doesn't quietly slip through unnoticed. Quadratics in particular have a lot of moving pieces — a discriminant, two possible roots, the possibility of complex numbers entirely — and getting a clear breakdown of each piece matters more than just getting a final answer.
Three Equation Types, Three Different Approaches
Linear Equations
ax + b = c → x = (c − b) ÷ a
This is the foundational case — one variable, no exponents, solved by isolating x through basic inverse operations. Move the constant to the other side, then divide by the coefficient.
Quadratic Equations
ax² + bx + c = 0 → x = (−b ± √(b²−4ac)) / 2a
The quadratic formula handles any equation with a squared term, and the ± symbol means there are typically two solutions rather than one. The exact number and type of solutions depends entirely on a single value hiding inside that formula: the discriminant.
Systems of Two Equations
a₁x + b₁y = c₁ and a₂x + b₂y = c₂, solved using Cramer's Rule
When two equations share two unknowns, you need both equations simultaneously to pin down exact values for x and y. Cramer's Rule uses a determinant-based approach to solve for both variables directly, without needing substitution or elimination by hand.
Understanding the Discriminant
The discriminant (Δ = b² − 4ac) is genuinely the most important number hiding inside the quadratic formula, because it tells you upfront what kind of answer you're about to get, before doing any further calculation:
- Δ > 0: two distinct real roots — the equation crosses the x-axis at two separate points
- Δ = 0: one repeated real root — the equation just barely touches the x-axis at a single point
- Δ < 0: no real roots — the equation never touches the x-axis at all, and the solutions become complex numbers
This tool computes the discriminant automatically and tells you exactly which case you're dealing with before showing the final roots, which genuinely saves a lot of confusion when a quadratic doesn't behave the way you expected.
Equation Solver — Equation Types
| Type | General Form |
|---|---|
| Linear | ax + b = 0 |
| Quadratic | ax² + bx + c = 0 |
| Cubic | ax³ + bx² + cx + d = 0 |
| System (2 variables) | a₁x + b₁y = c₁ ; a₂x + b₂y = c₂ |
| Exponential | aᵃ = b |
How to Use This Equation Solver
- Pick your equation type — Linear, Quadratic, or System 2×2 — from the tabs at the top
- For linear equations, enter values for a, b, and c following the form ax + b = c
- For quadratics, enter a, b, and c for ax² + bx + c = 0, and both roots appear along with the discriminant and root type
- For systems, enter coefficients for both equations separately, and x and y are solved simultaneously using Cramer's Rule
- Tap "Steps" on any tab to see the complete working, including formula substitution and a verification check against the original equation
Real Example: When Does a Ball Hit the Ground?
A ball is thrown upward, and its height over time follows h = −5t² + 20t + 1. The question: when does it hit the ground, meaning when does h equal zero?
Setting up the equation: −5t² + 20t + 1 = 0, so a=−5, b=20, c=1.
Discriminant: Δ = b² − 4ac = 400 − 4(−5)(1) = 400 − (−20) = 420.
Applying the quadratic formula: t = (−20 ± √420) / (−10), which gives t₁ ≈ 4.049 and t₂ ≈ −0.049.
Since negative time doesn't make physical sense in this context, the negative root gets rejected, leaving the ball hitting the ground at approximately 4.05 seconds. This is a great example of a case where the math produces two mathematically valid answers, but only one makes sense given the real-world context of the problem.
Equation Solver — Solved Examples
| Equation | Solution |
|---|---|
| x + 4 = 10 | x = 6 |
| x² − 4 = 0 | x = 2, −2 |
| 2x + y = 10, x − y = 2 | x = 4, y = 2 |
| x³ − 8 = 0 | x = 2 |
Why Systems of Equations Need Both Equations at Once
A single equation with two unknowns — like x + y = 10 — has infinite possible solutions. Adding a second equation constrains the problem down to exactly one solution (assuming the lines aren't parallel), which is exactly why systems require solving two equations together rather than one at a time. Cramer's Rule handles this using determinants: D = a₁b₂ − a₂b₁, then x = Dx/D and y = Dy/D, where Dx and Dy are calculated by substituting the constant terms into modified versions of that same determinant.
When a System Has No Unique Solution
If the main determinant D comes out to exactly zero, something specific is happening geometrically: the two lines represented by the equations are either parallel (no solution exists at all) or identical (infinitely many solutions exist). This calculator flags a zero determinant immediately with a clear error, rather than attempting to divide by zero and returning a meaningless result.
Where These Equation Types Actually Get Used
- Physics: projectile motion, exactly like the ball example, relies entirely on quadratic equations
- Finance: break-even analysis often reduces to solving a linear or quadratic equation
- Engineering: systems of equations describe circuits, structural forces, and material balances
- Business analytics: supply and demand intersections are classic system-of-equations problems
Connecting Back to Algebra
Solving an equation is really just a specific application of broader algebraic skills. If a problem needs an expression expanded or simplified before it's ready to solve — turning (x+5)² into x² + 10x + 25, for instance — the Algebra Solver handles that preparation step before this tool takes over to actually solve for the unknown.
Systems and Matrices Are the Same Idea in Different Clothes
A system of two equations can also be represented and solved as a matrix problem, and the underlying determinant math is genuinely identical either way. For larger systems, or for anyone who wants to see the matrix representation directly, the Matrix Calculator handles the same Cramer's Rule logic through matrix operations rather than a dedicated system-solving interface.
Verifying Results With Roots
Quadratic solutions often involve square roots directly in the formula, and double-checking that square root calculation independently is a good habit when working through a problem by hand. The Square Root Calculator is useful for that exact verification step, especially when a discriminant doesn't produce a clean perfect square.
A Mistake Worth Avoiding
A frequent error with the quadratic formula is forgetting that the ± applies to the entire square root term, not just part of it — some people accidentally compute only the addition case and miss the second root entirely. Both roots genuinely exist mathematically whenever the discriminant is positive, and skipping one means missing half the actual solution.
Beyond Basic Equation Solving
For equations that combine algebra with trigonometric or logarithmic functions, the Scientific Calculator and Log Calculator extend well past what linear and quadratic solving alone can cover. And for problems involving exponential growth that need solving for an unknown exponent rather than an unknown x, that's genuinely logarithm territory rather than standard equation solving.
Frequently Asked Questions
What does the discriminant actually tell you?
It tells you how many real solutions a quadratic equation has before you even calculate them. Positive means two real roots, zero means one repeated root, and negative means no real roots at all — just complex ones.
How do I solve a system of two equations with two unknowns?
Using Cramer's Rule: calculate the main determinant D from the coefficients, then two modified determinants Dx and Dy by substituting in the constant terms. x = Dx/D and y = Dy/D give the solution.
What happens if a quadratic has a negative discriminant?
The equation has no real solutions — the roots become complex numbers involving i, meaning the equation's graph never actually crosses the x-axis.
Can I enter a simple equation like 3x = 15?
Yes — enter a=3, b=0, c=15, since the form is ax + b = c. This solves to x = (15−0)/3 = 5.
Is this equation solver free to use?
Completely free, no sign up required, working instantly on any device.
Explore more calculators: Algebra Solver | Matrix Calculator | Square Root Calculator | Scientific Calculator
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