What Is the Quadratic Formula?
The quadratic formula is a method for solving any quadratic equation of the form ax² + bx + c = 0, where a is not zero. It tells you the values of x where the parabola crosses the x-axis — also known as the roots of the equation. A quadratic formula calculator applies this method instantly so you can check your homework or solve real-world problems without doing the arithmetic by hand.
How the Discriminant Determines the Roots
The discriminant, written as D = b² − 4ac, tells you what kind of roots the equation has without solving it completely. If D is positive, the equation has two distinct real roots. If D equals zero, there is one repeated real root, often called a double root. If D is negative, the equation has two complex conjugate roots that contain the imaginary unit i.
The Quadratic Formula Step by Step
The formula states that the solutions are given by:
First compute the discriminant b² − 4ac, then take its square root, and finally apply the plus and minus signs to find the two possible values of x. The calculator above shows each of these steps in the solution section.
Completing the Square vs. the Formula
Completing the square is another method for solving quadratics and is especially useful for rewriting an equation in vertex form. The quadratic formula is actually derived by completing the square on the general equation ax² + bx + c = 0. Both methods give the same answer, but the formula is faster for most equations with real coefficients.
The Vertex and the Parabola
Every quadratic equation graphs as a parabola. The vertex is its highest or lowest point, and its x-coordinate is always −b/(2a). The vertex lies exactly midway between the two roots when the equation has real solutions. Knowing the vertex helps you understand the maximum or minimum value of a real-world quantity such as profit or height.
FAQs
Can a quadratic equation have no solution?
It always has two solutions in the complex number system, but they are not real numbers when the discriminant is negative.
What happens if a equals zero?
The equation becomes linear, so the quadratic formula no longer applies. This calculator alerts you when that happens.
For more math help, try our fraction calculator, scientific calculator, or GPA calculator.