Quadratic Equation Solver
This quadratic equation solver finds the roots of any equation written as ax² + bx + c = 0. Just enter the three coefficients a, b, and c and the calculator returns both solutions, x₁ and x₂, updating live as you type. It also shows the discriminant, which tells you at a glance whether the equation has two real roots, one repeated root, or no real roots at all. It is ideal for algebra homework, physics problems, engineering, and any situation where you need to solve a second-degree equation quickly and accurately.
x₁
x₂
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Discriminant
Solves ax² + bx + c = 0. Results are rounded to 4 decimals.
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How it works
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. The solutions come from the quadratic formula x = (−b ± √(b²−4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant. When the discriminant is positive there are two distinct real roots; when it equals zero there is a single repeated real root; and when it is negative there are no real roots (the solutions are complex). The calculator computes the discriminant first, then applies the plus and minus branches of the formula to obtain x₁ and x₂. If a is zero the equation is no longer quadratic but linear (bx + c = 0), and the tool returns the single solution x = −c / b.
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Preguntas frecuentes
- What is a quadratic equation?
- A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. The highest power of the unknown is 2, which is what makes it quadratic. Such equations appear in geometry, projectile motion, optimization, and many other areas of mathematics and science.
- What is the quadratic formula?
- The quadratic formula gives the solutions of ax² + bx + c = 0 directly: x = (−b ± √(b²−4ac)) / 2a. The ± symbol means you compute two values, one using plus and one using minus, which give the two roots x₁ and x₂. This formula works for any quadratic equation, regardless of whether it can be factored easily.
- What is the discriminant and why does it matter?
- The discriminant is the expression b² − 4ac under the square root in the quadratic formula. Its sign tells you the nature of the roots without solving completely: if it is positive there are two distinct real roots, if it is zero there is one repeated real root, and if it is negative there are no real roots (the solutions are complex numbers).
- Can a quadratic equation have no real solutions?
- Yes. When the discriminant b² − 4ac is negative, the square root of a negative number is not a real value, so the equation has no real roots. For example, x² + x + 1 = 0 has a discriminant of 1 − 4 = −3, so it has no real solutions, only complex ones. This calculator reports that case explicitly.
- What happens if a equals zero?
- If a is zero the equation is not really quadratic, because the x² term disappears and it becomes the linear equation bx + c = 0. That has a single solution, x = −c / b (as long as b is not also zero). This calculator handles that case and returns the one root instead of two.