ax² + bx + c = 0
Solve ax² + bx + c = 0 using the quadratic formula
The quadratic formula is: x = (-b ± √(b²-4ac)) / 2a. This formula solves equations in the form ax² + bx + c = 0. The ± symbol means there are two solutions: one using + and one using -. The term under the square root (b²-4ac) is called the discriminant and determines the nature of the solutions.
Quadratic equations model real-world phenomena like projectile motion, profit optimization, area calculations, and physics problems. They're fundamental in algebra and appear in engineering, economics, and science. Understanding how to solve them is essential for higher mathematics and many technical fields.
The discriminant (b²-4ac) tells you about the solutions: Positive: Two distinct real solutions. Zero: One repeated real solution (the parabola touches the x-axis). Negative: Two complex solutions involving imaginary numbers (the parabola doesn't intersect the x-axis). This information helps graph the parabola and understand the equation's behavior.
Always write the equation in standard form (ax² + bx + c = 0) first. Check if factoring is possible before using the formula—it can be faster. If a=1 and b is even, completing the square may be easier. Verify your solutions by substituting them back into the original equation. Remember that real-world problems may require only positive solutions or solutions within a specific range.
The quadratic formula is x = (-b ± √(b²-4ac)) / 2a. It finds the solutions (roots) for any quadratic equation in the form ax² + bx + c = 0.
The discriminant is b²-4ac. If positive, there are two real solutions. If zero, there's one repeated solution. If negative, there are two complex solutions.
When the discriminant is negative, the equation has complex (imaginary) solutions. These involve the imaginary unit i = √(-1). Our calculator shows these as a ± bi.