Math

Equation Solver

The Equation Solver accepts any single-variable equation -- from simple linear expressions to transcendental functions involving sin, cos, and e^x -- and returns exact or numerical solutions with step-by-step explanations and an interactive graph showing roots as x-intercepts. It auto-detects the equation type, applies the appropriate solving method (quadratic formula, Cardano's method, Newton-Raphson, bisection), and displays both exact symbolic and decimal results.

Quick Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Quadratic Equation

Solve a classic quadratic with two real roots.

Key values: x^2 - 5x + 6 = 0 · Two real roots

Linear Equation

Solve a simple linear equation for x.

Key values: 3x + 7 = 22 · Single root

Transcendental Equation

Find roots of sin(x) = x/3 using numerical methods.

Key values: sin(x) = x/3 · Numerical solution

Cubic with Complex Roots

Solve a cubic equation and show complex roots.

Key values: x^3 + x + 1 = 0 · Complex roots enabled

Documentation

Documentation Contents

How to Use This Solver

Solve any single-variable equation in three simple steps.

Type your equation: Enter any equation in the input field. Use standard mathematical notation with $x$ as the variable. Examples: x^2 - 5x + 6 = 0, sin(x) = 0.5, e^x = 5.
Read the results: The solver auto-detects the equation type, selects the best solving method, and shows solutions instantly.
Explore the details: Expand the step-by-step panel to see the full solution process. Use the interactive graph to visualize the equation.

Tip

If you omit the = 0 part, the solver automatically assumes your expression equals zero. So typing x^2 - 9 is the same as typing x^2 - 9 = 0.

Supported Equation Types

The solver automatically detects your equation type and applies the most appropriate method.

Linear Equations

Form: $ax + b = 0$ . Always exactly one solution.

Example: 3x + 9 = 0 → x = -3

Quadratic Equations

Form: $ax^2 + bx + c = 0$ . Up to two solutions.

Example: x^2 - 5x + 6 = 0 → x = 2, 3

Cubic Equations

Form: $ax^3 + bx^2 + cx + d = 0$ . Up to three solutions.

Example: x^3 - 6x^2 + 11x - 6 = 0 → x = 1, 2, 3

Polynomial Equations

Degree 4 and above. Solved numerically (Newton-Raphson + bisection).

Example: x^5 - x - 1 = 0 → x ≈ 1.1673

Transcendental Equations

Equations involving sin, cos, tan, exp, log, sqrt. Solved numerically in a given domain.

Example: sin(x) = 0.5 → x ≈ 0.5236, 2.6180, ...

Key Formulas

Linear Equation

For $ax + b = 0$ where $a \neq 0$ :

x = -\frac{b}{a}

Quadratic Formula

For $ax^2 + bx + c = 0$ where $a \neq 0$ :

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the square root, $\Delta = b^2 - 4ac$ , is called the discriminant and determines the nature of the solutions.

Cardano's Formula (Cubic)

Cardano's 1545 formula solves cubics via a depressed cubic substitution. For the full derivation, discriminant analysis, and the casus irreducibilis, see the Cubic Equation Solver guide.

Newton-Raphson Method

The Newton-Raphson iteration $x_{n+1} = x_n - f(x_n)/f'(x_n)$ converges quadratically near a root. For convergence theory, failure cases, and geometric intuition, see the Newton-Raphson Calculator guide.

The Discriminant

For quadratic equations $ax^2 + bx + c = 0$ , the discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots:

Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two points. If Δ is a perfect square, the roots are rational.

Δ = 0: One repeated root (double root). The parabola touches the x-axis at exactly one point (its vertex).

Δ < 0: Two complex conjugate roots of the form $x = \frac{-b}{2a} \pm \frac{\sqrt{|\Delta|}}{2a}i$ . The parabola does not cross the x-axis.

Numerical Methods

When exact algebraic formulas are not available (degree 5 and above, or transcendental equations), the solver uses numerical methods:

Grid scan: The solver evaluates the function at 1,000 equally spaced points across the domain, looking for sign changes (places where the function crosses zero).
Bisection refinement: Each sign-change interval is refined using the bisection method, which halves the interval repeatedly until the root is located to high precision.
Newton-Raphson polishing: The bisection result is polished using Newton-Raphson iteration for even higher accuracy.

Why Numerical Methods?

The Abel-Ruffini theorem proves that there is no general algebraic formula for polynomial equations of degree 5 or higher. Transcendental equations (involving sin, cos, exp, log) generally have no closed-form solution either. Numerical methods are the only option.

Worked Examples

Example 1: Projectile Motion (Physics)

A ball is thrown upward from 5 feet with an initial velocity of 48 ft/s. The height equation is $h(t) = -16t^2 + 48t + 5$ . When does it hit the ground?

Setup: Set $h(t) = 0$ : -16t^2 + 48t + 5 = 0

Discriminant: $\Delta = 48^2 - 4(-16)(5) = 2304 + 320 = 2624$

Solutions: $t \approx -0.10$ (reject, negative time) and $t \approx 3.10$ seconds

Takeaway: In physics problems, always reject non-physical solutions (negative time, negative length).

Example 2: Chemical Equilibrium (Chemistry)

For a reaction with equilibrium constant $K = 0.04$ , the equilibrium equation becomes $0.04 = x^2 / (0.1 - x)^2$ .

Setup: Expand and rearrange to 0.96x^2 + 0.008x - 0.0004 = 0

Solution: $x \approx 0.0167$ (the equilibrium concentration shift)

Takeaway: Equilibrium problems commonly produce quadratic equations where only the positive root is physically meaningful.

Example 3: Break-Even Analysis (Engineering/Business)

Revenue $R(x) = 50x - 0.01x^2$ and cost $C(x) = 1000 + 30x$ . Find the break-even quantities.

Setup: Set $R(x) = C(x)$ : -0.01x^2 + 20x - 1000 = 0

Solutions: $x \approx 52.79$ and $x \approx 1947.21$ units

Takeaway: Break-even points occur where the cost and revenue curves intersect, which is the zero of their difference.

Example 4: Finding Intersection Points (Trigonometric)

Find where $\sin(x) = 0.5$ in the interval $[-10, 10]$ .

Method: Numerical scan with bisection + Newton-Raphson refinement

Solutions: $x \approx 0.5236, 2.6180$ and additional periodic roots in the domain

Takeaway: Transcendental equations may have infinitely many solutions. The solver shows those in the visible domain.

Common Mistakes

Forgetting to set one side to zero: The equation must be in the form $f(x) = 0$ before applying formulas. If you have $x^2 = 5x - 6$ , rewrite as $x^2 - 5x + 6 = 0$ .
Sign errors in the quadratic formula: When $b$ is negative, $-b$ becomes positive. Double-check your signs.
Confusing "no real solution" with "no solution": $x^2 + 4 = 0$ has no real solutions, but it does have two complex solutions: $x = \pm 2i$ .
Dividing by zero: Never divide both sides of an equation by an expression containing the variable. This can eliminate valid solutions.
Extraneous solutions from squaring: When you square both sides of an equation (e.g., to eliminate a square root), always verify solutions in the original equation.

Frequently Asked Questions

What does it mean when the solver says "no solution"?

A "no solution" (contradiction) result means no value of the variable satisfies the equation. For example, $0 = 5$ is always false. This is different from "no real solution," which means solutions exist in the complex numbers.

What are complex roots?

Complex roots involve the imaginary unit $i = \sqrt{-1}$ . They come in conjugate pairs (e.g., $2 + 3i$ and $2 - 3i$ ). Enable "Show complex roots" in Advanced Settings to see them.

Why does the solver use Newton-Raphson instead of an exact formula?

The Abel-Ruffini theorem proves that no general algebraic formula exists for polynomial equations of degree 5 or higher. Transcendental equations (sin, cos, exp, log) also lack closed-form solutions in general. Numerical methods are the only option.

Can this solver handle systems of equations?

This solver handles single-variable equations only. If you enter an equation with multiple variables (e.g., x + y = 5), it will notify you. For systems of equations, use a dedicated system solver.

How accurate are the numerical solutions?

Numerical solutions are accurate to approximately 10 decimal places (tolerance $10^{-10}$ ). The bisection method guarantees convergence, and Newton-Raphson provides quadratic convergence for polishing the result.

Is this solver free?

Yes, completely free with no paywalls or sign-ups. Unlike Symbolab and Mathway, we show step-by-step solutions at no cost.

Historical Background

The quest to solve equations spans over 4,000 years of mathematical history:

~2000 BCE: Babylonian scribes solved quadratic equations using geometric methods equivalent to completing the square, recorded on clay tablets.
~800 CE: Al-Khwarizmi wrote Al-jabr wa'l-muqabala, giving algebra its name and systematizing the solution of linear and quadratic equations.
1545: Gerolamo Cardano published Ars Magna, containing the general solution to cubic equations (attributed to del Ferro and Tartaglia) and the quartic solution by Ferrari.
1824: Niels Henrik Abel proved that no general algebraic formula exists for polynomial equations of degree 5 or higher (the Abel-Ruffini theorem).
1832: Evariste Galois developed group theory to explain precisely which equations are solvable by radicals and which are not.
1669-1690: Isaac Newton developed the Newton-Raphson method (refined by Joseph Raphson), providing the first systematic numerical approach to root-finding.

References

Wikipedia. "Quadratic formula." https://en.wikipedia.org/wiki/Quadratic_formula
Wikipedia. "Cubic equation." https://en.wikipedia.org/wiki/Cubic_equation
Wolfram MathWorld. "Quartic Equation." https://mathworld.wolfram.com/QuarticEquation.html
Wikipedia. "Newton's method." https://en.wikipedia.org/wiki/Newton%27s_method
Wikipedia. "Durand–Kerner method." https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method

Disclaimer

This equation solver is provided for educational and informational purposes only. While the solver uses mathematically rigorous algorithms, numerical methods may miss roots outside the scanned domain, and floating-point arithmetic can introduce small rounding errors. Always verify critical results independently. Do not rely on this tool for safety-critical engineering or medical calculations without independent verification by a qualified professional.