Equation solver
Set a Function Equal to a Target Variable
Use an explicit multiplication sign. Supported functions include sin, cos, tan, asin, acos, atan, sqrt, abs, ln, log, log10, exp, floor, and ceil.
Example Data
These sample settings show common function-to-target calculations.
| Function | Target | Bounds | Expected root | Suggested method |
|---|---|---|---|---|
| x^2 − 25 | 0 | 0 to 10 | 5 | Auto or Bisection |
| sin(x) | 0.5 | 0 to 2 | 0.52359878 | Newton-Raphson |
| exp(x) | 20 | 0 to 5 | 2.99573227 | Secant |
| sqrt(x + 4) | 5 | -4 to 30 | 21 | Auto |
Formula Used
The calculator rewrites the selected equation as a root problem.
Find x where g(x) = 0
Here, f(x) is your function and y is the target variable value. A root is valid when the residual, f(x) − y, is close to zero. Bisection repeatedly halves a bracket. Newton-Raphson uses xnext = x − g(x) / g′(x). Secant uses two recent points to estimate the slope without a direct derivative.
How to Use This Calculator
- Enter the function using the chosen single-letter variable.
- Enter the target value that the function must equal.
- Set lower and upper bounds around the expected solution.
- Choose auto for a general search or select a specific method.
- Set precision, iteration limits, and scan steps when needed.
- Press Solve Equation and verify the displayed substitution check.
- Download CSV or save a PDF record for later review.
Understanding Function-to-Variable Solving
A set function equal to variable calculator solves equations written as f(x) = y. The function may contain powers, roots, trigonometric terms, logarithms, and constants. The target y is a known output value. The calculator searches for an input x that makes both sides equal. This process is useful whenever an output is known before its input. Engineers use it for calibration. Students use it for algebra practice. Analysts use it for threshold checks. A reliable result requires a clear function, sensible bounds, and an appropriate method.
The expression box accepts normal mathematical notation. Use x as the unknown variable. Examples include x^2 - 9, sin(x), log(x) + 2, and sqrt(x + 4). Parentheses control the order of operations. The calculator also accepts constants such as pi and e. Choose a target of zero to solve ordinary root equations. Choose another target to find where a curve reaches a stated value. For example, setting x^2 equal to 25 returns values near -5 and 5. A selected interval determines which valid solution is reported.
Bounds make the calculation safer and more useful. They define the region that the solver examines. A wide interval can locate several possible sign changes. A narrow interval focuses the search on one expected solution. The automatic method scans the chosen range first. It then uses a bracketing process to refine the closest suitable root. Bisection works best when the function changes sign across the bounds. It is stable, though it can need more iterations. Newton-Raphson is usually faster near a smooth root. It estimates a slope numerically. Secant solving avoids direct derivative calculations and can work well with two good starting values.
Precision controls the stopping rule. More digits require a smaller tolerance. This can increase iterations, especially for flat curves. The maximum-iteration setting prevents endless attempts when a solution is difficult. Review the residual after every calculation. A residual close to zero means the substituted x value produces an output close to the target. The result panel also shows the method used, the selected root, the residual, and the elapsed iteration count. The history table helps identify steady convergence, oscillation, or a poor starting point.
Always check the equation after solving. Substitute the reported root into the original function. Compare the calculated function value with the target. Small differences can occur because decimal computers use rounded values. Avoid intervals containing discontinuities, such as division by zero or tangent asymptotes. Logarithms and square roots also require valid input domains. When several roots exist, try separate intervals around each expected crossing. Export the iteration record as CSV when you need a calculation trail. Use the print option to save a clean PDF report. These checks turn a numerical estimate into a dependable working answer.
Use degrees only after conversion, because trigonometric functions expect radians. Keep units consistent when targets represent physical measurements or observed data from experiments.
Frequently Asked Questions
What does this calculator solve?
It finds a variable value that makes your function equal the chosen target. Internally, it solves the equivalent equation f(x) − target = 0 within your selected bounds.
Which expressions can I enter?
Use numbers, one variable, parentheses, +, -, *, /, and ^. You can also use pi, e, sin, cos, tan, asin, acos, atan, sqrt, abs, ln, log, log10, exp, floor, and ceil.
Why should I use explicit multiplication?
Explicit multiplication avoids ambiguity. Write 2*x instead of 2x, and write 3*(x + 1) instead of 3(x + 1). This keeps the expression parser safe and predictable.
What is the target value?
The target is the output that your function must reach. For example, enter target 20 with exp(x) to solve exp(x) = 20. Use target zero for ordinary root finding.
When should I choose auto?
Choose auto when you know a sensible search interval but do not know the best method. It scans for sign changes, selects a nearby candidate, and then refines it with bisection.
Why does bisection require opposite signs?
Bisection relies on a continuous function crossing zero. If values at the bounds have opposite signs, at least one crossing is usually bracketed. Discontinuities can still create misleading sign changes.
When is Newton-Raphson useful?
Newton-Raphson is useful near a smooth root with a reasonable initial guess. It often converges quickly, but it can fail near flat slopes, discontinuities, or when a step leaves the selected interval.
Why did the solver report no root?
The interval may not contain a crossing, the scan may be too coarse, or the function may be undefined there. Try wider bounds, more scan steps, a new initial guess, or a different method.
Can one equation have several roots?
Yes. A curve can reach the same target many times. Auto displays detected candidates from scanned sign changes. Use tighter, separate intervals when you need a specific root.
Are trigonometric values in degrees?
No. All trigonometric functions use radians. Convert degrees first by multiplying degrees by pi/180. For example, sin(pi/6) equals 0.5.
How should I verify a reported answer?
Substitute the reported variable value into the original function. Compare that result with the target and review the residual. A small residual supports the numerical solution, subject to rounding and domain limits.