Equation Solver
Solve it and see why. Linear, quadratic, cubic, 2×2 and 3×3 systems, plus a numeric solver for equations that have no tidy formula.
ax + b = c
Solution
ax² + bx + c = 0
ax³ + bx² + cx + d = 0
Two equations, two unknowns
The second row reads minus — enter a negative number there to make it a plus.
Solution
Three equations, three unknowns
Solution
Solve anything for x
Angles are read in radians here, which is the usual convention for equations of this kind.
How to use this calculator
Step by step
- Pick the shape of your equation along the top. If it does not match any of them, use Any equation — that mode solves numerically and handles almost anything.
- Fill in the coefficients in the equation line. The equation is written out in full so you can see exactly which number goes where, including the signs.
- The answer updates as you type. Below it, the steps panel shows the substitutions that produced it — that is the part to copy into your homework.
- For quadratics and cubics, the plot underneath shows where the curve meets the x-axis, which is a fast way to sanity-check the roots.
- In Any equation, type both sides separately (for example
cos(x)andx) and give a search range. Every crossing inside that range is reported.
Good to know
- Signs matter: to solve x² − 5x + 6 = 0, enter b as −5. The equation line shows a plus, so a negative coefficient reads as a subtraction.
- If your equation is not set to zero, move everything to one side first. 2x² = 3x + 5 becomes 2x² − 3x − 5 = 0.
- The discriminant tells you the answer before you compute it: positive means two crossings, zero means the curve just touches, negative means it never reaches the axis.
- The numeric solver only finds roots inside the range you give it. If you expect more solutions — trigonometric equations have infinitely many — widen the range and look for the pattern.
- Roots found numerically come with a residual: how far from zero the equation actually is at that point. A residual near 1e-15 is exact as far as the arithmetic can tell.
The maths behind it
- Quadratic formula x = ( −b ± √(b² − 4ac) ) ÷ 2a The discriminant b² − 4ac decides how many real roots exist.
- Vertex x = −b ÷ 2a The turning point, and the axis of symmetry of the parabola.
- Cramer’s rule (2 × 2) x = (ce − bf) ÷ (ae − bd), y = (af − cd) ÷ (ae − bd) A zero determinant means the lines are parallel or identical.
- Cardano’s substitution x = t − b ÷ 3a → t³ + pt + q = 0 Removes the squared term so a cubic can be solved in closed form.
- Bisection if f(a) and f(b) have opposite signs, a root lies between them Halve the interval repeatedly; it converges every time.
What do complex roots mean?
They mean the parabola never touches the x-axis, so there is no real number that satisfies the equation. The roots still exist as complex numbers of the form a ± bi, and they always come in conjugate pairs when the coefficients are real.
Why does the 3 × 3 solver sometimes say “no unique solution”?
Because the three planes described by your equations do not meet at a single point. Either two are parallel (no solution at all) or all three share a line (infinitely many solutions). The steps panel names the column where elimination stalled.
Can it solve for something other than x?
The numeric mode always calls the unknown x, but the letter is arbitrary — rename your variable to x and the answer is the same.
How accurate are the numeric roots?
Bisection runs 80 halvings, which drives the bracket well below the precision of double-precision arithmetic. The reported residual shows how close to zero the equation actually is at the root.