Equation Solver

Quick Answer

Last updated: 10 November 2026 ·How we calculate this: exact algebra for linear equations and bisection for the rest

How This Equation Solver Works

Type an equation in x with one equals sign. If the equation is linear (x appears only to the first power), the solver moves every term to one side, collects them, and divides, showing each step and the exact answer. For anything else, such as x² − 5x + 6 = 0, 2^x = 10 or sin(x) = 0.5, it rewrites the equation as f(x) = 0, scans the search range for places where f(x) changes sign, and narrows each one down to the solution.

How to Use the Equation Solver

1. Type your equation using x, for example 2x + 3 = 11 or x^2 – 5x + 6 = 0.
2. Set the search range if you expect answers outside -100 to 100.
3. Press Calculate.
4. Read the solutions and the check that both sides match.
5. Follow the steps to see how the answer was found.

How Equations Are Solved

• Linear: ax + b = cx + d becomes (a − c)x = d − b, so x = (d − b) ÷ (a − c)
• Same operation on both sides: adding, subtracting, multiplying or dividing both sides by the same non-zero number keeps the equation true
• Root finding: if f(a) and f(b) have opposite signs and f is continuous, there is a root between a and b (Intermediate Value Theorem)
• Bisection: check the midpoint, keep the half with the sign change, and repeat

Key Terms

• Equation: a statement that two expressions are equal.
• Solution or root: a value of x that makes the equation true.
• Linear equation: an equation where x appears only to the first power.
• Identity: an equation that is true for every value of x.
• Bisection: a method that finds a root by repeatedly halving an interval where the sign changes.

Worked Example

Solve 4(x − 2) = 2x + 6.

• Expand: 4x − 8 = 2x + 6
• Subtract 2x from both sides: 2x − 8 = 6
• Add 8 to both sides: 2x = 14
• Divide by 2: x = 7. Check: 4(7 − 2) = 20 and 2(7) + 6 = 20.

Real-Life Examples

• Phone bill: 20 + 0.1x = 35 gives x = 150 minutes.
• Rectangle: x(x + 3) = 40 gives x = 5 (and x = −8, which a length cannot be).
• Doubling time: 2^x = 10 gives x ≈ 3.321928.
• Trig: sin(x) = 0.5 between 0 and 6.3 gives x ≈ 0.523599 and x ≈ 2.617994 (in radians).

Equation vs Expression

• An expression such as 3x + 5 has no equals sign. You can simplify it but not solve it.
• An equation such as 3x + 5 = 20 says two expressions are equal, so you can solve for x.
• An inequality such as 3x + 5 > 20 has a range of answers; use the Inequality Calculator.

Examples of Equations and Solutions

When an Equation Has No Solution

Some equations have no real solution. x² = −4 has no real answer because no real number squared is negative. 2x + 3 = 2x + 5 has no solution because the x terms cancel and leave 3 = 5. On the other hand, 2(x + 1) = 2x + 2 is true for every x, an identity. The solver tells you which case you have. For numeric solutions, it only searches the range you set, so widen the range if you expect a very large or very small answer.

Equations in Everyday Problems

Equations turn word problems into something you can solve. A phone plan costs $20 a month plus $0.10 a minute; how many minutes make the bill $35? Write 20 + 0.1x = 35 and solve to get 150 minutes. A rectangle is 3 m longer than it is wide and has an area of 40 m²; write x(x + 3) = 40 and solve to get a width of 5 m. Interest, speed, mixtures and break-even problems all work the same way.

Common Mistakes to Avoid

• Doing something to only one side. Whatever you add, subtract, multiply or divide, do it to both sides.
• Dropping a negative sign when moving a term across the equals sign.
• Dividing by an expression that can be zero. You may lose a solution; factor instead.
• Writing 2x^2 as (2x)^2. The power applies only to x unless you add brackets.
• Searching too small a range. Widen it if you expect large answers.

Frequently Asked Questions

References & Sources