Equation Solver
Solve linear, quadratic, cubic, and polynomial equations with step-by-step solutions. Enter values for instant results with step-by-step formulas.
Formula
Quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
The quadratic formula solves ax^2 + bx + c = 0. For linear equations ax + b = 0, x = -b/a. For cubic equations, the Rational Root Theorem and synthetic division reduce the problem to a quadratic. The discriminant (b^2 - 4ac) determines whether roots are real or complex.
Worked Examples
Example 1: Solve a Quadratic Equation
Problem: Solve x^2 - 5x + 6 = 0
Solution: Using the quadratic formula: a=1, b=-5, c=6\nDiscriminant = (-5)^2 - 4(1)(6) = 25 - 24 = 1\nSince discriminant > 0, two real roots exist\nx = (5 + sqrt(1)) / 2 = 6/2 = 3\nx = (5 - sqrt(1)) / 2 = 4/2 = 2\nVerification: 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0\nVerification: 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0
Result: x = 3 and x = 2 (sum of roots = 5, product = 6)
Example 2: Solve a Cubic Equation
Problem: Solve x^3 - 6x^2 + 11x - 6 = 0
Solution: Try rational roots (factors of 6): x=1, x=2, x=3\nTest x=1: 1 - 6 + 11 - 6 = 0 (root found!)\nSynthetic division by (x-1): x^2 - 5x + 6 = 0\nSolve quadratic: discriminant = 25 - 24 = 1\nx = (5+1)/2 = 3 and x = (5-1)/2 = 2\nAll three roots: x = 1, x = 2, x = 3
Result: x = 1, x = 2, and x = 3
Frequently Asked Questions
What is a linear equation and how do I solve it?
A linear equation is an equation where the variable appears only to the first power, taking the general form ax + b = 0 where a and b are constants. To solve a linear equation, isolate the variable x by performing inverse operations on both sides. Subtract b from both sides to get ax = -b, then divide both sides by a to get x = -b/a. Linear equations always have exactly one solution unless a equals zero, in which case the equation is either an identity (true for all x when b also equals zero) or a contradiction (no solution when b is not zero). Linear equations represent straight lines when graphed and are the foundation for more complex equation types.
What does the discriminant tell us about an equation?
The discriminant is the expression under the square root in the quadratic formula, calculated as b squared minus 4ac. It reveals crucial information about the nature and number of solutions without actually solving the equation. A positive discriminant means the equation has two distinct real roots and the parabola crosses the x-axis at two points. A discriminant of zero means there is exactly one real root (a repeated root) and the parabola touches the x-axis at its vertex. A negative discriminant means there are no real roots, only two complex conjugate roots, and the parabola does not intersect the x-axis at all. The discriminant is also useful for analyzing conic sections and higher-degree polynomials.
How do I solve a cubic equation?
Solving cubic equations of the form ax cubed plus bx squared plus cx plus d equals zero typically involves finding one rational root using the Rational Root Theorem, then reducing to a quadratic equation through synthetic division or polynomial long division. The Rational Root Theorem states that any rational root p over q must have p as a factor of the constant term d and q as a factor of the leading coefficient a. Once you find one root, divide the cubic by the factor (x minus that root) to get a quadratic, which you solve using the quadratic formula. If no rational roots exist, numerical methods like Newton-Raphson iteration or Cardano formula for the depressed cubic can find the roots.
What is the difference between an equation and an expression?
An equation contains an equals sign and states that two expressions are equal, such as 2x plus 3 equals 7, and it can be solved to find specific values of the variable that make the statement true. An expression is a mathematical phrase without an equals sign, such as 2x plus 3, and it cannot be solved but can be simplified or evaluated for specific values of x. Equations have solutions (specific values), while expressions have values that depend on the variable. You can simplify expressions by combining like terms and applying operations, but solving requires an equation with two sides. Equation Solver works with equations, finding the values of x that make the left side equal to zero.
Can I share or bookmark my calculation?
You can bookmark the calculator page in your browser. Many calculators also display a shareable result summary you can copy. The page URL stays the same so returning to it will bring you back to the same tool.
Can I use Equation Solver on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.