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Quadratic Formula Solver

Solve quadratic equations in the form ax² + bx + c = 0 and find the real or complex roots.

Enter Equation Coefficients

What is the Quadratic Formula?

The quadratic formula is a powerful mathematical tool used to find the roots (or solutions) of a quadratic equation, which is a polynomial equation of the second degree. Any equation that can be written in the form ax² + bx + c = 0, where 'a' is not zero, can be solved using this formula. This makes it a universally applicable method, unlike factoring, which only works for specific equations.

The Formula and the Discriminant

The roots of the quadratic equation are given by the formula:

x = [-b ± √(b² - 4ac)] / 2a

The term inside the square root, b² - 4ac, is known as the discriminant (Δ). The value of the discriminant is critical because it tells us the nature of the roots without having to fully solve the equation:

  • If Δ > 0, there are two distinct real roots. This means the parabola of the equation crosses the x-axis at two different points.
  • If Δ = 0, there is exactly one real root (also called a repeated or double root). The vertex of the parabola touches the x-axis at a single point.
  • If Δ < 0, there are two distinct complex roots, which involve imaginary numbers. The parabola does not cross the x-axis at all. Our calculator can handle these complex solutions.

Applications of Quadratic Equations

Quadratic equations are essential in many fields of science, engineering, and finance. They can model situations where a quantity increases and then decreases, or vice versa. For example:

  • Physics: Calculating the trajectory of a thrown object (projectile motion) under the influence of gravity.
  • Engineering: Designing curved structures like bridges or satellite dishes. The shape of a suspension bridge's main cable is a parabola.
  • Finance: Finding the maximum revenue or profit for a business, which is often modeled by a parabolic curve.
  • Geometry: Problems involving area can lead to quadratic equations. For example, finding the dimensions of a rectangle with a given area and perimeter. This connects to tools like our Area Calculator. For a deeper understanding of algebraic concepts, resources like Khan Academy offer excellent tutorials.