Equation Solver
Solve linear, quadratic, and simultaneous equations with step-by-step solutions and interactive graphs
Coefficients
Solution
Step-by-Step Solution
About this tool
The equation solver is a free tool for students that returns the solutions and the worked steps for linear, quadratic, and simultaneous equations. For quadratics it also shows the discriminant, the vertex, and the parabola graph, helping you understand the quadratic formula visually. It is great for checking homework answers, reviewing solution methods, and testing your understanding before an exam.
How to use
- 1 Choose the type of equation to solve (linear, quadratic, or simultaneous).
- 2 Enter the coefficients of each term into the fields.
- 3 The solutions and intermediate steps are displayed automatically.
- 4 For quadratics, also review the discriminant, vertex, and graph.
How it works
A linear equation ax + b = 0 has the solution x = -b / a, the point where the line crosses the x-axis. A quadratic ax2 + bx + c = 0 is solved with the quadratic formula x = (-b +/- sqrt(b2 - 4ac)) / 2a. The term b2 - 4ac is the discriminant, and its sign decides the number of solutions: positive gives two distinct real roots, zero gives one repeated root, and negative gives no real roots but two complex ones. The vertex of the parabola has x-coordinate -b / 2a. A simultaneous system finds where two lines intersect, solved by substitution or elimination. If the lines are parallel there is no solution; if they coincide there are infinitely many.
Frequently asked questions
What happens when the discriminant is negative?
There are no real solutions; the parabola does not cross the x-axis. Over the complex numbers, two solutions still exist.
When does a simultaneous system have no solution?
When the two lines are parallel and never meet. Conversely, if both equations describe the same line, there are infinitely many solutions.
Why does the quadratic formula work?
It is derived by completing the square on ax2 + bx + c = 0. Because this tool shows the steps, you can follow the transformation and understand it.
What does the vertex represent?
It is the lowest or highest point of the parabola, with x-coordinate -b / 2a, and it matters for maximum and minimum problems and the axis of symmetry.
Related tools and uses
To see the shape of your equation, the graph plotter is handy. For word problems with percentages, see the percentage calculator, and for bases of coefficients try the number base converter.