Quadratic Formula Calculator

For ax² + bx + c = 0 with a not equal to 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The ± sign gives two values, one with a plus and one with a minus. The formula works for every quadratic equation, whether or not it can be factored.

Write the equation as ax² + bx + c = 0, then read a, b and c with their signs. Work out b² − 4ac, take its square root, and compute (−b + root) / (2a) and (−b − root) / (2a). Enter a, b and c here and the calculator shows each step.

The discriminant is b² − 4ac, the expression under the square root in the quadratic formula. If it is positive, there are two distinct real solutions. If it is 0, there is one repeated real solution. If it is negative, there are no real solutions, only two complex ones.

The equation has no real solutions, and the parabola never crosses the x-axis. It does have two complex solutions, (−b ± i√(−Δ)) / (2a), written with the imaginary number i, where i² = −1. For x² + x + 1 = 0, Δ = −3 and the solutions are −0.5 ± 0.866025i. The result sentence gives them.

Then the equation is not quadratic. It becomes bx + c = 0, with the single solution x = −c / b when b is not 0. The quadratic formula would divide by 2a = 0, so the calculator asks for a value of a other than 0.

Move all terms to one side so the other side is 0, and put the terms in the order x², x, number. For 2x² = 3x + 5, subtract 3x and 5 from both sides to get 2x² − 3x − 5 = 0, so a = 2, b = −3 and c = −5. Then the discriminant is 9 + 40 = 49 and x = 2.5 or −1.

Here a = 1, b = −5 and c = 6. The discriminant is 25 − 24 = 1, so x = (5 ± 1) / 2, which gives x = 3 and x = 2. You can check it by factoring: (x − 2)(x − 3) = x² − 5x + 6.

It is a solution that counts twice, which happens when the discriminant is 0. For x² − 4x + 4 = 0, Δ = 16 − 16 = 0 and the only solution is x = −b / (2a) = 2. On the graph, the parabola touches the x-axis at its vertex without crossing it.

The x-coordinate of the vertex is −b / (2a), halfway between the two solutions, and the y-coordinate is c − b² / (4a). For x² − 5x + 6, the vertex is at x = 2.5 and y = 6 − 25 / 4 = −0.25. This calculator gives the solutions, not the vertex, but these formulas use the same a, b and c.

Put each solution back into the equation. For x² − 5x + 6 = 0, x = 2 gives 4 − 10 + 6 = 0 and x = 3 gives 9 − 15 + 6 = 0, so both are correct. You can also check that the solutions add up to −b / a and multiply to c / a: 2 + 3 = 5 and 2 · 3 = 6.

Factoring is quick when the discriminant is a perfect square, because the solutions are then rational numbers, as in x² − 5x + 6. The quadratic formula always works, including when the solutions are irrational or complex, and it is the safe choice when factoring is not obvious. To factor x² + bx + c, look for two numbers that multiply to c and add up to b, such as −2 and −3 for x² − 5x + 6, which gives (x − 2)(x − 3). The factor calculator lists the factor pairs of a whole number.

So the order does not depend on the sign of a. Whether the plus sign or the minus sign in the formula gives the smaller value changes with the sign of a: for x² − 5x + 6 the minus sign gives 2, but for −x² + 4x − 3 it gives 3. The calculator sorts the values, so x₁ is never larger than x₂.

It comes from completing the square. Multiplying ax² + bx + c = 0 by 4a gives 4a²x² + 4abx + 4ac = 0, and adding b² − 4ac to both sides gives 4a²x² + 4abx + b² = b² − 4ac, which is (2ax + b)² = b² − 4ac. Taking the square root of both sides and solving for x gives x = (−b ± √(b² − 4ac)) / (2a).

They describe thrown objects and falling objects, where height depends on time squared, and they show up in area problems, such as finding the sides of a rectangle from its perimeter and area. A quadratic also models profit when a price change affects both the price and the number sold.