Quadratic Formula Calculator
Solve ax² + bx + c = 0 for its real roots.
Solve any quadratic equation ax² + bx + c = 0 for its real roots.
How the Math Works
The Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0 by applying the universal formula x = (-b ± √(b² - 4ac)) ྊ 2a. The process begins by identifying the coefficients a, b, and c from your equation, then calculating the discriminant (b² - 4ac) which determines the nature of the roots. If the discriminant is positive, you'll get two distinct real solutions; if zero, one repeated real solution; and if negative, the roots are complex numbers. The calculator handles the arithmetic automatically, showing both possible solutions from the ± operator in the formula.
Practical Applications
To use this calculator for practical problems, first rearrange your equation into standard form ax² + bx + c = 0, ensuring all terms are on one side and the equation equals zero. Identify your coefficients: a is the multiplier of x², b is the multiplier of x, and c is the constant term. Enter these values into the respective fields and compute. This tool is invaluable for checking manual calculations, solving physics problems involving projectile motion, finding break-even points in business models, or determining dimensions in geometric problems where quadratic relationships occur.
Day-to-Day Use
Beyond academic exercises, quadratic equations appear in everyday scenarios like calculating the optimal price for maximum profit, determining how long a thrown object will take to hit the ground, or figuring out the dimensions needed for a rectangular garden with a fixed area. The calculator saves time when you need quick answers without working through the discriminant and arithmetic manually. Whether you're a student checking homework, an engineer verifying calculations, or a curious individual solving practical optimization problems, this tool provides instant, reliable solutions to quadratic equations you'll encounter in work or personal projects.
Worked example
x² − 3x + 2 = 0 → disc = 1, roots x = 2 and x = 1.
FAQ
What if the discriminant is negative?
There are no real roots — the solutions are complex numbers.