Loona Calculators
Solve linear equations (ax + b = cx + d) and quadratic equations (ax² + bx + c = 0) with every step shown — including the discriminant, repeated and complex roots, and a final check that substitutes the answer back in.
Linear equations in the form ax + b = cx + d, and quadratic equations in the form ax² + bx + c = 0. It shows every step, the discriminant for quadratics, and a final check. It does not solve cubic equations, simultaneous equations in two unknowns, or inequalities — the page title says exactly what it covers.
Sometimes the x terms cancel out completely. If the numbers also match (like 5x + 1 = 5x + 1) the equation is always true — infinitely many solutions. If the numbers do not match (5x + 1 = 5x + 3) it can never be true — no solution. The solver tells you honestly which case you have instead of inventing an answer.
When a quadratic's discriminant (b² − 4ac) is negative, the graph never crosses the x-axis and there are no real answers. The roots are then complex numbers using i, where i² = −1 — for example, x² + 1 = 0 has roots x = i and x = −i. This solver shows them and still checks them.
Linear equations are solved by keeping the equation balanced — whatever you do to one side, you do to the other. First the x terms are gathered on the left (subtract cx from both sides), then the number terms on the right (subtract b), and finally both sides are divided by the coefficient of x.
Quadratic equations use the quadratic formula. For ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) ÷ (2a)
The part under the root, D = b² − 4ac, is called the discriminant, and it tells you the kind of answer before you finish: D > 0 means two different real roots, D = 0 means one repeated root, and D < 0 means two complex roots and no real answer.
Enter 0 for the missing coefficient. x² − 9 = 0 becomes a = 1, b = 0, c = −9, and the solver gives x = 3 or x = −3. For a linear equation like 2x = 10, that is a = 2, b = 0, c = 0, d = 10.
Then the x² term disappears and the equation is really linear. The solver notices this and solves it as a linear equation instead, telling you that is what happened.
Substituting the answer back into the original equation is the surest way to catch a mistake: if both sides come out equal, the answer is certainly right. It is a good habit for homework too — teachers often give method marks for a visible check.
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