Frequently Asked Questions

A quadratic equation is a second-degree polynomial equation in the form \(ax^2 + bx + c = 0\), where \(a \neq 0\). The highest power of x is 2, which is why it's called "quadratic" (from "quadratus" meaning square).

A quadratic equation always has exactly two solutions. However, these solutions can be:

• Two distinct real roots (when \(\Delta > 0\))
• One double root (when \(\Delta = 0\))
• Two complex roots (when \(\Delta < 0\))

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It provides the solutions to any quadratic equation \(ax^2 + bx + c = 0\) and works for all cases, including complex roots.

Factoring is fastest when the quadratic has integer coefficients and factors nicely. Use it when you can quickly find two numbers that multiply to ac and sum to b. Try our Factoring Calculator.

Use the quadratic formula when factoring doesn't work easily, when coefficients are not integers, or when you need exact solutions. It's always reliable and never fails.

Completing the square transforms a quadratic into vertex form \(a(x-h)^2 + k\). It's useful for graphing, finding vertices, and deriving the quadratic formula. See our Completing the Square Calculator.

The discriminant is \(\Delta = b^2 - 4ac\). It tells you the nature of the roots without solving the equation:

• \(\Delta > 0\): Two distinct real roots
• \(\Delta = 0\): One double root
• \(\Delta < 0\): Two complex roots

The discriminant tells you about the graph of the quadratic: if it crosses the x-axis (2 roots), touches it (1 root), or doesn't touch it (no real roots). Try our Discriminant Calculator.

Quadratic equations appear in many real-world applications:

• Physics: Projectile motion, object dropped from height
• Business: Profit maximization, cost functions
• Engineering: Parabolic arches, satellite dishes
• Sports: Ball trajectories

Vertex form \(a(x-h)^2 + k\) directly reveals:

• Vertex at (h, k)
• Direction (opens up if a>0, down if a<0)
• Maximum/minimum value