Equation solver
Type an equation with both sides; get the steps, every root and a check by substitution.
One unknown, any single letter. Use ^ for powers, brackets, fractions like 1/3 and decimals like 0.25: 3x + 5 = 20, x^2 - 5x + 6 = 0, x^3 - 6x^2 + 11x - 6 = 0. Polynomial equations up to degree 8.
How do I solve a linear equation?
Expand the brackets, collect the terms with the unknown on one side and the numbers on the other, then divide by the coefficient. For 4(x − 3) + 12 = 15 − 5(x + 6): expand to 4x − 12 + 12 = 15 − 5x − 30, simplify to 4x = −15 − 5x, move the x terms to get 9x = −15, and divide: x = −15/9 = −5/3 (OpenStax Algebra and Trigonometry 2e, section 2.2, Example 2). The calculator shows each of these steps for the equation you type and then substitutes the answer back into both sides to confirm they are equal.
What if there is no solution, or every number is a solution?
When the unknown cancels, only a statement about numbers is left. 2x + 3 = 2x + 5 becomes 3 = 5, which is never true, so there is no solution. 2x + 3 = 3 + 2x becomes 3 = 3, which is true for every x, so every number is a solution (an identity). The same happens with higher powers: x² = x² + 4 has no solution because the squares cancel and 0 = 4 is left.
How are quadratic equations solved?
The calculator moves everything to one side, clears fractions, and finds a, b and c. It computes the discriminant D = b² − 4ac. If D is a perfect square the quadratic factors over the whole numbers and you get two rational roots; if D = 0 there is one repeated root; if D > 0 but not a square the roots are irrational and shown with a radical; if D < 0 they are complex and shown with i. Examples: x² − 3x − 5 = 0 gives x = (3 ± √29)/2 (OpenStax, section 2.5, Example 8), x² + x + 2 = 0 gives x = (−1 ± i√7)/2 (Example 10), and 4x² + 15x + 9 = 0 factors as (4x + 3)(x + 3) = 0 with roots −3/4 and −3 (Example 4).
How are equations of degree 3 to 8 solved?
First the rational zero theorem: if a polynomial with whole-number coefficients has a rational root p/q, then p divides the constant term and q divides the leading coefficient. The calculator lists these candidates, tests each one exactly (not with rounded decimals) and divides out every factor it finds. What remains is a quadratic, solved exactly as above, or a polynomial with no rational roots, which is solved numerically: the number of real roots is confirmed exactly with Sturm’s theorem, every real root is bracketed by a sign change of the polynomial computed in exact arithmetic, and each complex root is checked by substituting it back. Example: for x³ − 6x² − x + 30 = 0 the roots are −2, 3 and 5 (OpenStax, section 5.5, Example 2), and for 4x³ − 3x − 1 = 0 they are 1 and −1/2 twice (Example 5).
What equations can it not solve?
Equations with the unknown in a denominator (rational equations), under a root, in an exponent or inside sin, cos, ln and the like; inequalities; equations with more than one unknown; and polynomials of degree above 8. For a rational equation, multiply both sides by the common denominator first, type the polynomial equation that results, and check at the end that no answer makes an original denominator zero (OpenStax, section 2.2). For several unknowns use the system of equations calculator.
How is this different from the other equation calculators?
The quadratic formula calculator and the cubic equation calculator take the coefficients a, b, c (and d) one by one. The system of equations calculator solves several linear equations with several unknowns. The completing the square calculator shows that one method for a quadratic, and the polynomial calculator adds, multiplies and divides polynomials without solving anything. Here you type the whole equation as you would write it, with brackets and an unknown on both sides, and it picks the right method for the degree, up to degree 8, and verifies every root.
How do I type an equation?
Write both sides with one equals sign, for example 2(x - 3) = x/2 + 1 or x^3 - 6x^2 + 11x - 6 = 0. Use ^ for powers (x² also works), brackets freely, fractions such as 1/3 or x/2, and decimals such as 0.25; products can be written 2x or 2*x. Any single letter can be the unknown. Fractions and decimals are read exactly, so 0.5x + 1/3 = 2 gives x = 10/3, not a rounded decimal.
What mistakes should I watch for when solving equations by hand?
Dividing both sides by an expression that contains the unknown loses a solution: from x² = 5x, dividing by x gives x = 5 and drops x = 0, so bring everything to one side and factor instead (x² − 5x = 0 gives x = 0 and x = 5). Sign errors when moving terms and when distributing a minus sign are the next most common, which is why the steps spell out each operation. Finally, always substitute the answers back, especially after multiplying by an expression with the unknown or squaring, which can create extraneous solutions. The calculator substitutes every root into both sides for you.
Sources: OpenStax, Algebra and Trigonometry 2e: section 2.2 “Linear Equations in One Variable”, section 2.5 “Quadratic Equations”, section 2.6 “Other Types of Equations” and section 5.5 “Zeros of Polynomial Functions” (rational zero theorem; worked examples used as automated tests). Related: quadratic formula calculator, cubic equation calculator, system of equations calculator, completing the square calculator, polynomial calculator, synthetic division calculator and factor calculator.