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➗ MathFree
Multiplicative persistence calculator
Multiply a number's digits again and again until one digit is left: 39 becomes 27, then 14, then 4, so its multiplicative persistence is 3.
A
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➗ MathFree
Depressed cubic solver
t³ − 3t + 2 = 0 has the roots 1, 1 and −2, since it factors as (t − 1)²(t + 2): enter p and q and get all three roots with the discriminant and Cardano's…
A
Runs in your browser
➗ MathFree
Inverse variation calculator
If x is 4 when y is 6, the constant of variation is 24, so an x of 8 gives a y of 3, with each substitution step shown.
K
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➗ MathFree
Linear equation from slope and point
Slope 2 through the point (1, 3) gives y − 3 = 2(x − 1), which simplifies to y equals 2x + 1 because the y-intercept is 3 − 2 × 1, that is 1.
A
Runs in your browser
➗ MathFree
Literal equation solver
Solve y = mx + b for x and get x = (y − b) / m, with the steps that undo each operation. No numbers are needed: the result stays a formula you can reuse.
F
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➗ MathFree
Solve quadratic by factoring
Factor a quadratic the way your textbook does: x² − 5x + 6 becomes (x − 2)(x − 3), so the roots are 2 and 3.
P
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➗ MathFree
Quadratic with complex roots calculator
Get exact complex roots, not rounded decimals: x² + x + 1 has discriminant −3, so its roots are −1/2 + (√3/2)i and −1/2 − (√3/2)i, a conjugate pair.
K
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➗ MathFree
Vertex form to standard form calculator
Turn a vertex-form parabola into standard form without expanding by hand: 2(x − 3)² − 5 becomes 2x² − 12x + 13, using b = −2ah and c = ah² + k.
B
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➗ MathFree
Quadratic in form solver
Crack a biquadratic equation in steps: with u in place of x², x⁴ − 5x² + 4 = 0 becomes u² − 5u + 4 = 0, giving 1 and 4 for u and four real solutions, −2, −1, 1…
F
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➗ MathFree
Parabola x-intercepts from vertex form
Get the x-intercepts straight from vertex form, without expanding: y = (x − 2)² − 4 crosses the x-axis at 0 and 4, each 2 units from the vertex at (2, −4).
P
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➗ MathFree
Cube equation solver
x³ = 8 has three solutions, not one: the real root 2 and the complex pair -1 ± i√3. Enter any real k and get all three roots of x³ = k as real and imaginary…
S
Runs in your browser
➗ MathFree
Sum and product of roots calculator
For x^2 - 5x + 6 = 0 the roots add up to 5 and multiply to 6, read straight from the coefficients with Vieta's formulas: the sum is −b/a and the product is c/a.
S
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➗ MathFree
Two-step linear equation solver
6x + 5 = 20 becomes 6x = 15 and then gives 5/2 for x, kept as an exact fraction. The solver removes the constant, divides by the coefficient and checks the…
B
Runs in your browser
➗ MathFree
Y-intercept of a line calculator
The line 2x − y + 6 = 0 crosses the y-axis at (0, 6): enter the three coefficients of the general form and get the intercept, or a clear notice when the line is…
S
Runs in your browser
➗ MathFree
Casting out nines check
Casting out nines checks a sum or product in seconds: a claimed total of 579 leaves remainder 3 mod 9, the same as its addends’ remainders 6 and 6 combined, so…
S
Runs in your browser
➗ MathFree
Imaginary quadratic class number calculator
Enter a squarefree d and get the exact class number of Q(√−d): for 23 the discriminant is −23 and the class number is 3, while 1 gives the Gaussian rationals…
K
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➗ MathFree
Fibonacci coding converter
Turn any positive integer into its Fibonacci code: 6 becomes 10011, built from the summands 1 and 5.
P
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➗ MathFree
Hofstadter Q sequence
In Hofstadter’s Q sequence the 10th term is 6 and the 20th is 12. Each new term looks back by distances that earlier terms decide, which makes the growth…
S
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➗ MathFree
Narayana's cows sequence calculator
Get any term of Narayana's cows sequence from index 0 to 97: N(10) is 28, because each term adds the one just before it to the one three places back, 19 plus 9.
B
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➗ MathFree
p-adic valuation calculator
Count how many times a prime divides a whole number: 72 is 8 times 9, so it contains the prime 2 three times, its 2-adic valuation is 3 and the leftover part is…
P
Runs in your browser
➗ MathFree
Perrin sequence term
Get any term of the Perrin sequence, which starts 3, 0, 2 and adds the terms two and three places back: P(10) is 17, that is 10 plus 7.
K
Runs in your browser
➗ MathFree
Thue-Morse sequence bit
Position 7 is 111 in binary, three ones, so its Thue-Morse bit is 1. Enter any non-negative position to see its binary form, how many ones it has and the…
A
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➗ MathFree
Lower Wythoff sequence calculator
The 10th term of the lower Wythoff sequence is 16, and the sequence opens 1, 3, 4, 6, 8. Enter a position counted from one to get that exact term without…
F
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➗ MathFree
Absolute value inequality solver
|2x − 3| < 5 holds for −1 < x < 4, the interval (−1, 4), while a greater-than version gives two outer rays instead.
P
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➗ MathFree
Cubic inequality solver
x³ − 2x² − 5x + 6 ≥ 0 holds on [−2, 1] ∪ [3, ∞): enter the four coefficients and the comparison, and see the real roots, the sign on every interval and the…
B
Runs in your browser
➗ MathFree
2x2 system elimination method calculator
Solve two linear equations exactly by elimination: for 2x + 3y = 7 paired with x minus y equal to 1, x cancels out, 5y = 5 remains, and the point is (2, 1).
F
Runs in your browser
➗ MathFree
Line and parabola intersection
The line y = x + 2 meets the parabola y = x^2 at (-1, 1) and (2, 4): the calculator sets both equal, solves the quadratic and reads the discriminant, here 9, to…
A
Runs in your browser
➗ MathFree
Two-variable linear inequality calculator
For 2x + y < 6 you get a dashed boundary where 2x + y equals 6, the half-plane below y = −2x + 6, and the origin as a test point that works.
S
Runs in your browser
➗ MathFree
Factor difference of cubes calculator
Check that two whole numbers are perfect cubes and factor their difference: 125 − 8 becomes (5 − 2)(25 + 10 + 4), because 125 is 5³ and 8 is 2³.
B
Runs in your browser
➗ MathFree
Homogeneous linear system 2x2 calculator
Enter the four coefficients of two equations with zero on the right and get the full solution set: with 1, 2, 3 and 4 the determinant is −2, so only x and y…
K
Runs in your browser
➗ MathFree
Polynomial composition calculator
Substitute one polynomial into another and get the expanded result: with f(x) = x² and g(x) = 1 + x, f(g(x)) is 1 + 2x + x², of degree 2. Each step is shown.
S
Runs in your browser
➗ MathFree
Polynomial from roots calculator
Turn known roots into a polynomial in standard form: roots 1, 2 and 3 give (x − 1)(x − 2)(x − 3), which expands to x³ − 6x² + 11x − 6.
P
Runs in your browser
➗ MathFree
Polynomial subtraction calculator
Subtract polynomials column by column: 4x^3 − 2x + 7 minus x^3 + 5x^2 − 3 gives 3x^3 − 5x^2 − 2x + 10, with every power lined up so you can check each step.
F
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➗ MathFree
FOIL method calculator
Multiply two binomials and see every FOIL product in order: (2x + 3)(4x − 5) gives 8x², −10x, 12x and −15, so the result is 8x² + 2x − 15.
K
Runs in your browser
➗ MathFree
Inverse function calculator
Reverse a function and see each step: f(x)=2x+3 has the inverse f⁻¹(x)=0.5(x − 3), valid for all real numbers.
B
Runs in your browser
➗ MathFree
Polynomial coefficients extractor
Paste a polynomial and get its coefficients in order, from the highest power down to the constant: 3x^2 + 2x − 5 gives 3, 2 and −5, and x^3 − 4x gives 1, 0, −4…
S
Runs in your browser
➗ MathFree
Polynomial degree calculator
Read the degree of a polynomial straight off its coefficient list: 3, −2, 0 and 5 describe 3 − 2x + 5x³, a cubic of degree 3 whose leading coefficient is 5.
A
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➗ MathFree
Polynomial standard form calculator
Tidy a messy polynomial in one step: 3x^2 + 4x − 7 + 2x^3 − x^2 + 5 − 4x becomes 2x^3 + 2x^2 − 2, of degree 3. Fractions and decimals stay exact.
F
Runs in your browser
➗ MathFree
Difference quotient calculator
Type the coefficients of a polynomial and see [f(x+h)−f(x)]/h already simplified: for f(x) = x² it becomes 2x + h, and letting h go to zero leaves the…
A
Runs in your browser
➗ MathFree
Function evaluation calculator
Pick a function type, enter its coefficients and a value of x, and get f(x) at once: f(x)=2x+1 evaluated at 3 gives 7.
P
Runs in your browser
➗ MathFree
Even or Odd Function Checker
Enter a polynomial and find out whether it is even, odd or neither: 3x⁴ − 2x² + 7 is even, because replacing x with −x leaves it unchanged.
F
Runs in your browser
➗ MathFree
Function reflection calculator
Mirror a function across either axis and see the new rule with its points: across the y-axis, x² − 4x + 3 becomes (−x)² − 4(−x) + 3 and the point (2, −1) moves…
B
Runs in your browser
➗ MathFree
Vertical asymptote calculator for rational functions
(x + 1)/(x^2 - x - 2) has a single vertical asymptote, at 2, because the factor for -1 cancels and leaves only a hole.
K
Runs in your browser
➗ MathFree
Square root function domain calculator
√(x - 4) is defined only for x ≥ 4, so its domain is [4, ∞) and the graph starts at 4 on the x-axis.
S
Runs in your browser
➗ MathFree
Common logarithm calculator
The common logarithm of 1000 is 3, because ten cubed is 1000: enter any positive number and get the power of ten behind it, negative for values between 0 and 1.
F
Runs in your browser
➗ MathFree
Doubling Time Calculator
Enter a growth rate per period and see how long it takes to double: at 5% per period, growth compounded once per period doubles in 14.2066990829 periods, worked…
B
Runs in your browser
➗ MathFree
Function iteration calculator
Feed a rule its own output step after step and see the whole orbit: x² − 1 starting at 1.5 reaches −0.68359375 after three steps and −0.532699584960938 after…
K
Runs in your browser
➗ MathFree
Logarithm quotient rule calculator
Turn log(x/y) into log x − log y in one step, with the quotient rule spelled out. When the base and both values are numbers, the difference is also evaluated…
A
Runs in your browser
➗ MathFree
Logarithmic to exponential form converter
Enter base 2, argument 8 and value 3 to see log₂(8) = 3 rewritten as 2³ = 8: the base stays the base, the value becomes the exponent and the argument becomes…
S
Runs in your browser
➗ MathFree
One-to-one function checker
Find out whether a function has an inverse before you try to build one: f(x) = 2x − 1 passes the horizontal line test, because every output comes from exactly…
P
Runs in your browser
➗ MathFree
Complex number addition calculator
(3 + 2i) + (1 − 4i) is 4 − 2i: enter the real and imaginary parts of two complex numbers and get their sum in standard form, with the rule shown next to the…
P
Runs in your browser
➗ MathFree
Complex modulus calculator
The modulus of 3 + 4i is 5, its distance from the origin on the complex plane: enter the real and imaginary parts and get |z| and its square, never negative.
S
Runs in your browser
➗ MathFree
Complex number power calculator
(1 + i)² is 2i: enter the real and imaginary parts and a whole exponent, and get the power in a + bi form with the modulus and angle behind it, no long…
A
Runs in your browser
➗ MathFree
Complex Square Root Calculator
The square roots of 3 + 4i are 2 + i and −2 − i: enter the real and imaginary parts and get the principal root and its counterpart in rectangular form, ready to…
B
Runs in your browser
➗ MathFree
Logarithmic inequality solver
Solve log₂(x − 1) > 3 and get x > 9, or (9, ∞): the tool checks that x − 1 stays positive, rewrites the inequality as x − 1 > 8 and flips the sign when the base…
K
Runs in your browser
➗ MathFree
Simplify logarithmic expressions
Type log(100x²/y) + log(y) and get 2 + 2 log(x): the logs are split by the standard rules, the log of 100 becomes exactly 2, and the opposite log(y) terms…
F
Runs in your browser
➗ MathFree
Number of terms in an arithmetic sequence
From 3 to 27 in steps of 4 there are 7 terms: enter the first term, the last term and the common difference, and get the count n with both ends included.
A
Runs in your browser
➗ MathFree
Product of complex conjugates calculator
(3 + 4i) times its conjugate (3 − 4i) is 25: enter the real and imaginary parts and get a² + b², the squared modulus of the number, always real and never…
P
Runs in your browser
➗ MathFree
Complex cube roots calculator
The three cube roots of 8 are 2, −1 + i√3 and −1 − i√3: enter any complex number and get all three roots, 120° apart, in polar and rectangular form, with angles…
F
Runs in your browser
➗ MathFree
Imaginary part of a complex number calculator
The imaginary part of 7 − 4i is −4: enter a number as a + bi or by its size and angle, and get the signed coefficient of i, ready to compare, graph or reuse.
K
Runs in your browser
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