Quadratic Equation Calculator
Enter the coefficients a, b, and c of a quadratic equation (ax² + bx + c = 0) to calculate its roots.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real-number coefficients and a is not equal to zero. The term "quadratic" comes from the Latin word quadratus, meaning "square," because the variable is raised to the second power.
Quadratic equations appear throughout mathematics, physics, engineering, economics, and everyday problem-solving. They model projectile motion, area optimization, profit maximization, and many other scenarios where a relationship involves a squared variable.
The Quadratic Formula
The quadratic formula provides a direct way to find the solutions (roots) of any quadratic equation. It is derived by completing the square on the general form and works for all cases -- real roots, repeated roots, and complex roots.
The expression under the square root, b² - 4ac, is called the discriminant and it determines the nature of the roots.
The Discriminant and Number of Solutions
The discriminant is the key to understanding how many solutions a quadratic equation has and what type they are:
- If b² - 4ac > 0: The equation has two distinct real roots. The parabola crosses the x-axis at two points.
- If b² - 4ac = 0: The equation has one repeated real root (also called a double root). The parabola touches the x-axis at exactly one point (the vertex).
- If b² - 4ac < 0: The equation has two complex conjugate roots. The parabola does not cross the x-axis at all.
Checking the discriminant before solving can save time and tell you what kind of answer to expect.
Factoring vs. the Quadratic Formula
When possible, factoring is often the fastest method for solving a quadratic equation. Factoring works well when the roots are integers or simple fractions. However, many quadratic equations do not factor neatly, and in those cases the quadratic formula is the reliable fallback that always works.
Quadratic equations can also be expressed in vertex form: a(x - h)² + k = 0, where (h, k) is the vertex of the parabola. This form is especially useful for graphing and identifying the maximum or minimum value of the quadratic function.
Worked Example: Solving x² - 5x + 6 = 0
Let us solve x² - 5x + 6 = 0 using both factoring and the quadratic formula.
Method 1 -- Factoring:
Find two numbers that multiply to 6 and add to -5: those are -2 and -3.
x² - 5x + 6 = (x - 2)(x - 3) = 0
Therefore x = 2 or x = 3
Method 2 -- Quadratic Formula:
a = 1, b = -5, c = 6
Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
x = (5 ± sqrt(1)) / 2 = (5 ± 1) / 2
x = 6/2 = 3 or x = 4/2 = 2
Both methods yield the same roots: x = 2 and x = 3. The discriminant equals 1 (positive), confirming two distinct real roots.