Calculator Use
The quadratic formula calculator is an essential tool for students, engineers, and mathematicians designed to solve quadratic equations of the form ax² + bx + c = 0. By entering the three coefficients (a, b, and c), this tool provides the roots of the equation instantly, including complex numbers if the discriminant is negative.
Whether you are checking homework or solving complex physics trajectories, this quadratic formula calculator ensures accuracy and provides step-by-step logic to help you understand the derivation of the answer.
- Coefficient (a)
- The number multiplying the squared term (x²). This value cannot be zero.
- Coefficient (b)
- The number multiplying the linear term (x). If x is missing, b is zero.
- Constant (c)
- The standalone number without a variable. If no constant exists, c is zero.
How It Works
The calculator utilizes the standard quadratic formula, which is derived from completing the square of a general quadratic equation. The formula is:
x = [-b ± √(b² – 4ac)] / 2a
The calculation process involves three primary components:
- The Discriminant (b² – 4ac): This value tells us the nature of the roots. If positive, there are two real roots. If zero, there is one real root. If negative, there are two complex roots.
- The ± Symbol: This indicates that we perform two separate calculations—one using addition and one using subtraction—to find the two potential values of x.
- The Denominator (2a): The entire numerator is divided by twice the value of 'a' to finalize the result.
Calculation Example
Example: Solve the equation 2x² + 8x – 10 = 0 using the quadratic formula calculator.
Step-by-step solution:
- Identify coefficients: a = 2, b = 8, c = -10.
- Calculate the discriminant: D = 8² – 4(2)(-10) = 64 + 80 = 144.
- Find the square root of the discriminant: √144 = 12.
- Apply the formula for x₁: (-8 + 12) / (2 * 2) = 4 / 4 = 1.
- Apply the formula for x₂: (-8 – 12) / (2 * 2) = -20 / 4 = -5.
- Result: x = 1, x = -5.
Common Questions
Can 'a' be zero?
No. If 'a' is zero, the x² term disappears and the equation becomes linear (bx + c = 0). The quadratic formula involves division by 2a, so if a = 0, the equation is undefined in the context of quadratic math.
What are complex roots?
Complex roots occur when the discriminant (the part under the square root) is negative. Since you cannot take the square root of a negative number in the real number system, we use the imaginary unit i (where i = √-1). This calculator automatically handles these cases for you.
Why use the formula instead of factoring?
Factoring only works easily when the roots are integers or simple fractions. The quadratic formula calculator is superior because it works for every single quadratic equation, regardless of how messy the decimals or irrational numbers may be.