Quadratic Formula Calculator
ax² + bx + c = 0
This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax² + bx + c = 0 for x, where a ≠ 0, using the quadratic formula.
The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant (b² − 4ac) is less than, greater than or equal to 0.
- When b² − 4ac = 0 there is one real root.
- When b² − 4ac > 0 there are two real roots.
- When b² − 4ac < 0 there are two complex roots.
A Surprising Story: From Classroom to Cosmos
An astrophysics student was struggling with calculating the trajectory of a satellite. The path was described by a complex quadratic equation, and a small error in calculation could mean the satellite would be lost in space. Frustrated after hours of manual calculation, she remembered the quadratic formula calculator she used in school.
She entered the complex coefficients into the calculator. Instantly, it provided the two possible time coordinates for the satellite's position. But the surprise was in the details. The calculator also showed the discriminant was positive, confirming two real, viable points in its trajectory. This simple, instant verification gave her the confidence she needed. The same formula that helps students pass algebra exams is also used to navigate the cosmos, proving that fundamental mathematics is truly the language of the universe. It's a surprising reminder that the tools we learn early on can have applications beyond our wildest dreams.
Quadratic Formula:
The quadratic formula is used to solve quadratic equations where a ≠ 0:
Examples using the quadratic formula
Example 1: Two Real Roots (x² − 8x + 5 = 0)
Here, a = 1, b = -8, c = 5. The discriminant b²-4ac = (-8)² - 4(1)(5) = 64 - 20 = 44 > 0, so there are two real roots.
x = (8 ± √44) / 2 = 4 ± √11
x ≈ 7.31662 and x ≈ 0.683375
Example 2: Two Complex Roots (5x² + 20x + 32 = 0)
Here, a = 5, b = 20, c = 32. The discriminant b²-4ac = 20² - 4(5)(32) = 400 - 640 = -240 < 0, so there are two complex roots.
x = (-20 ± √-240) / 10 = (-20 ± i√240) / 10 = -2 ± (2i√15)/5
x ≈ -2 + 1.54919i and x ≈ -2 - 1.54919i
FAQs
What is a quadratic equation?
A quadratic equation is a second-order polynomial equation in a single variable x with the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.
What is the discriminant?
The discriminant is the part of the quadratic formula under the square root sign: b² − 4ac. It is important because it determines the number and type of roots the equation has: one real, two real, or two complex roots.