Quadratic Equation Solver
Solve quadratic equations in the form ax² + bx + c = 0
Coefficient of x²
Coefficient of x
Constant term
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0.
Discriminant: The discriminant (b² - 4ac) determines the nature of the solutions:
- If discriminant > 0: Two distinct real solutions
- If discriminant = 0: One repeated real solution
- If discriminant < 0: No real solutions (complex roots)
Formula: x = (-b ± √(b² - 4ac)) / (2a)
What does the discriminant tell us about solutions?
The discriminant (b^2 - 4ac) reveals the nature of solutions without solving: positive means two distinct real roots, zero means one repeated real root (the parabola touches the x-axis), and negative means two complex conjugate roots (the parabola doesn't cross the x-axis). The discriminant's value also indicates how far apart the roots are.
Why can't the coefficient 'a' be zero?
If a=0, the equation becomes bx + c = 0, which is linear, not quadratic. The x^2 term is what makes an equation quadratic (from Latin 'quadratus' meaning square). A true quadratic always has a parabolic graph, while a linear equation graphs as a straight line.
What is completing the square and how does it relate to the quadratic formula?
Completing the square transforms ax^2 + bx + c into a(x-h)^2 + k form, revealing the vertex at (h,k). The quadratic formula is derived by completing the square on the general form: x = (-b +/- sqrt(b^2-4ac))/(2a). This technique is also fundamental in deriving the equations of conic sections.
Where are quadratic equations used in real life?
Quadratic equations model projectile motion (throwing a ball), calculate areas, optimize profits in business, design parabolic mirrors and satellite dishes, determine braking distances, and analyze electrical circuits. Any scenario involving acceleration, maximization/minimization, or squared relationships typically involves quadratics.