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Equation Solver

Solve linear and quadratic equations.

Equation type
Quadratic: a·x² + b·x + c = 0

Auto-detect treats it as quadratic when a ≠ 0, otherwise linear (b·x + c = 0).

Quadratic equation

x₁ = 2, x₂ = 1

  1. Equation: 1x² + -3x + 2 = 0
  2. Discriminant D = b² − 4ac = -3² − 4·1·2 = 1
  3. D > 0 → two distinct real roots.
  4. x = (−b ± √D) / 2a = (3 ± 1) / 2
  5. x₁ = 2
  6. x₂ = 1
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How to use Equation Solver

What this tool does

The equation solver works out the solutions to linear equations of the form ax + b = c and quadratic equations of the form ax² + bx + c = 0. You enter the coefficients, choose the equation type or let the tool detect it, and it returns the answer along with a clear, line-by-line account of how it got there. It loads with a worked quadratic — x² − 3x + 2 = 0 — so the layout and the style of the steps are visible immediately.

For quadratics the tool always shows the discriminant and names the case — two real roots, one repeated root, or a complex conjugate pair — so you learn why the answer looks the way it does, not just what it is. For linear equations it shows the subtract-then-divide steps, and it correctly handles the edge cases where there is no solution or every value works.

When you would use it

Students reach for it to check homework: solve the equation by hand, then confirm the roots and compare your working against the steps shown. It is equally useful the other way round — if your answer disagrees, the step list points to where the two paths diverge. Teachers can use it to generate clean worked examples, since the discriminant line and the method labels mirror how quadratics are taught.

Outside the classroom, linear and quadratic equations turn up whenever you solve for an unknown in a formula: rearranging a pricing or break-even equation, finding when a thrown object reaches a height, or back-solving a physics relationship. The solver gives you the value quickly and shows the reasoning so you can trust it.

How to use it

  1. Choose Linear, Quadratic, or leave it on Auto-detect.
  2. Type the coefficients. For a linear equation ax + b = c, fill a, b and c. For a quadratic ax² + bx + c = 0, fill a, b and c.
  3. Press Solve.
  4. Read the headline answer, then the numbered steps below it — for a quadratic that includes the discriminant and the substitution into the quadratic formula.
  5. Use Copy solution to copy the answer and the full working as text.

How to read the result

A linear result is a single value of x. A quadratic result is one of three shapes: two distinct values when the discriminant is positive, a single repeated value when it is exactly zero, or a complex pair written as p + qi and p − qi when it is negative. The steps spell out the method: for a quadratic, the discriminant b² − 4ac is computed first, then the quadratic formula x = (−b ± √D) / 2a is applied.

If you only need a parabola’s roots without the linear case, the dedicated quadratic equation solver is a focused alternative. To see the equation as a curve, plot it with the graph plotter, and to work a simple linear equation out one explained step at a time, use the algebra step-by-step tool.

Privacy

Every part of this tool runs in your browser. The coefficients you type are processed by JavaScript on your own device — there is no upload, no logging and no storage between visits. Refreshing the page restores the sample equation. Because nothing depends on a server, the solver keeps working offline once the page has loaded.

Frequently asked questions

What is the difference between a linear and a quadratic equation?
A linear equation has the variable only to the first power — ax + b = c — and its graph is a straight line, so it has exactly one solution (unless a is zero). A quadratic equation has the variable squared — ax² + bx + c = 0 — and its graph is a parabola, so it can have two solutions, one repeated solution, or two complex solutions. This tool handles both: choose the type yourself, or let Auto-detect pick based on whether the squared coefficient a is non-zero.
What is the discriminant and why does it matter?
The discriminant is the quantity D = b² − 4ac inside the square root of the quadratic formula. Its sign tells you the nature of the roots before you finish solving. If D is positive there are two distinct real roots; if D is exactly zero there is one repeated real root where the parabola just touches the x-axis; if D is negative the roots are a pair of complex conjugates. The tool shows D and states which case applies in its working.
What does it mean when the answer has an i in it?
The letter i is the imaginary unit, defined so that i² equals −1. When a quadratic has a negative discriminant it has no real solutions, but it still has two complex solutions of the form p + qi and p − qi. These come up in physics and engineering and are perfectly valid answers — they simply mean the parabola never crosses the x-axis. The tool reports the real part and the imaginary part separately.
Can an equation have no solution or infinitely many?
Yes, for linear equations. If you enter coefficients where a is zero, the equation ax + b = c collapses to b = c. If that is true the equation holds for every x — infinitely many solutions — and if it is false it holds for no x at all. The tool detects both of these and explains them rather than dividing by zero. Quadratics with a non-zero a always have exactly two roots, counting a repeated root twice and counting complex roots.
Is my equation sent anywhere when I solve it?
No. The coefficients you type and the whole solving process stay in your browser. The algebra runs in JavaScript on your device, nothing is uploaded to a server, and nothing is saved between visits. Refreshing the page resets the tool to its sample equation. You can solve homework or work problems here knowing the inputs never leave your computer.

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