Q: What is the discriminant of a quadratic equation?

The discriminant (delta) is the value b² - 4ac. It determines the nature of the solutions: if delta > 0 there are two distinct real solutions, if delta = 0 there are two equal solutions, if delta < 0 the solutions are complex conjugates (containing the imaginary unit i).

Q: What is the quadratic formula?

The quadratic formula is x = (-b +/- sqrt(delta)) / 2a, where delta = b² - 4ac. The +/- sign indicates two solutions: x1 with + and x2 with -. This formula was developed by Arab mathematicians in the 9th century.

Q: What are complex solutions?

When the discriminant is negative, the solutions contain the imaginary unit i (where i² = -1). They always come in conjugate pairs: a + bi and a - bi. They have important applications in electrical engineering, quantum mechanics, and signal theory.

Quadratic Equation Calculator

Solve quadratic equations ax² + bx + c = 0. Calculate discriminant, real and complex solutions.

How to Solve a Quadratic Equation

This calculator solves equations in the form ax² + bx + c = 0 using the quadratic formula, showing the discriminant, solutions, and result type (real or complex).

The Quadratic Formula

The formula for finding solutions of a quadratic equation is:

x = (-b +/- sqrt(delta)) / 2a

where delta = b² - 4ac is the discriminant. This formula was formalized by the Indian mathematician Brahmagupta in the 7th century and refined by Arab mathematicians, particularly al-Khwarizmi in the 9th century.

The Role of the Discriminant

The discriminant delta completely determines the nature of the solutions:

delta > 0: two distinct real solutions. The parabola intersects the x-axis at two points.
delta = 0: two equal real solutions (a double root). The parabola is tangent to the x-axis.
delta < 0: two complex conjugate solutions. The parabola does not intersect the x-axis.

Vieta's Formulas

The solutions x1 and x2 are connected to the coefficients by elegant relationships, discovered by French mathematician Francois Viete in the 16th century:

x1 + x2 = -b/a (sum of solutions)
x1 x x2 = c/a (product of solutions)

These relationships are useful for verifying results and solving inverse problems.

Practical Applications

Quadratic equations appear in many contexts: projectile motion in physics, profit maximization in economics, curve intersections in geometry, and arch design in engineering.

Frequently Asked Questions

What is the discriminant of a quadratic equation?▼

The discriminant (delta) is the value b² - 4ac. It determines the nature of the solutions: if delta > 0 there are two distinct real solutions, if delta = 0 there are two equal solutions, if delta < 0 the solutions are complex conjugates (containing the imaginary unit i).

What is the quadratic formula?▼

The quadratic formula is x = (-b +/- sqrt(delta)) / 2a, where delta = b² - 4ac. The +/- sign indicates two solutions: x1 with + and x2 with -. This formula was developed by Arab mathematicians in the 9th century.

What are complex solutions?▼

When the discriminant is negative, the solutions contain the imaginary unit i (where i² = -1). They always come in conjugate pairs: a + bi and a - bi. They have important applications in electrical engineering, quantum mechanics, and signal theory.