Calculator Use
Our quadratic equation calculator is a powerful tool designed to solve second-degree polynomial equations in the form of ax² + bx + c = 0. Whether you are dealing with real numbers or complex imaginary results, this calculator provides the exact roots (x-intercepts) and the vertex of the parabola. It is an essential resource for students, engineers, and math enthusiasts who need a reliable way to verify algebraic solutions.
To use this tool, simply input the coefficients of your equation into the designated fields. The calculator automatically computes the discriminant to determine the nature of the roots before applying the quadratic formula.
- Coefficient (a)
- The number preceding the x² term. This cannot be zero, as that would make the equation linear rather than quadratic. It determines how steeply the parabola opens.
- Coefficient (b)
- The number preceding the x term. If there is no x term in your equation, enter zero.
- Constant (c)
- The numerical value without a variable. This represents the y-intercept of the parabola.
How It Works
The quadratic equation calculator uses the standard Quadratic Formula to find the values of x. The process involves several distinct mathematical steps to ensure accuracy, especially when dealing with negative discriminants.
x = [-b ± sqrt(b² – 4ac)] / 2a
- Discriminant (D = b² – 4ac): This value tells us how many solutions exist. If D > 0, there are two real roots. If D = 0, there is one real root. If D < 0, the roots are complex.
- The Vertex: The turning point of the parabola is calculated using x = -b/2a.
- Complex Numbers: When the discriminant is negative, the calculator utilizes 'i' (the square root of -1) to provide imaginary root solutions.
Calculation Example
Example Scenario: Solve the quadratic equation 2x² – 4x – 6 = 0.
Step-by-step solution using the calculator logic:
- Identify Coefficients: a = 2, b = -4, c = -6.
- Calculate Discriminant: D = (-4)² – 4(2)(-6) = 16 + 48 = 64.
- Solve for x₁: x₁ = (-(-4) + sqrt(64)) / (2 * 2) = (4 + 8) / 4 = 3.
- Solve for x₂: x₂ = (-(-4) – sqrt(64)) / (2 * 2) = (4 – 8) / 4 = -1.
- Result: The roots are 3 and -1. The parabola opens upward because a > 0.
Common Questions
What happens if the discriminant is zero?
When the discriminant (b² – 4ac) equals zero, it means the vertex of the parabola touches the x-axis at exactly one point. In this case, there is only one unique real solution, often called a double root or a repeated root.
Can this calculator solve equations with complex roots?
Yes. Our quadratic equation calculator is programmed to handle negative discriminants. It will provide the result in the standard complex form (a + bi), where 'a' is the real part and 'bi' is the imaginary part.
Why can't 'a' be zero?
If 'a' were zero, the x² term would disappear, leaving bx + c = 0. This is a linear equation, not a quadratic one. Linear equations are solved differently and do not follow the parabolic shape or require the quadratic formula.