Smallest Root Calculator
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Reviewed by Calculator Editorial Team
The smallest root calculator helps you find the smaller of the two solutions to a quadratic equation of the form ax² + bx + c = 0. This tool is useful in physics, engineering, and mathematics when you need to determine the minimum value of a quadratic function.
What is the smallest root of a quadratic equation?
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants. The roots of the equation are the values of x that satisfy the equation.
For a quadratic equation, there are two roots (which may be real or complex). The smallest root is the smaller of these two values. In real-world applications, the smallest root often represents a minimum value or critical point in a physical system.
Key Points
- The smallest root is the smaller of the two solutions to ax² + bx + c = 0
- It can be found using the quadratic formula
- For real roots, the discriminant (b² - 4ac) must be non-negative
How to find the smallest root
To find the smallest root of a quadratic equation, follow these steps:
- Identify the coefficients a, b, and c in the equation ax² + bx + c = 0
- Calculate the discriminant using the formula b² - 4ac
- If the discriminant is negative, there are no real roots
- If the discriminant is zero, there is one real root (x = -b/(2a))
- If the discriminant is positive, calculate both roots using the quadratic formula
- Select the smaller of the two roots as the smallest root
For complex roots, the smallest root is typically considered to be the one with the smaller imaginary part when both roots are complex.
Formula for smallest root
The roots of a quadratic equation ax² + bx + c = 0 can be found using the quadratic formula:
Quadratic Formula
x = [-b ± √(b² - 4ac)] / (2a)
The smallest root is the one with the minus sign in the numerator:
Smallest Root Formula
xsmallest = [-b - √(b² - 4ac)] / (2a)
This formula gives the smaller of the two real roots when the discriminant is positive.
Example calculation
Let's find the smallest root of the equation 2x² - 5x + 2 = 0.
- Identify coefficients: a = 2, b = -5, c = 2
- Calculate discriminant: (-5)² - 4(2)(2) = 25 - 16 = 9
- Since discriminant is positive, there are two real roots
- Calculate roots using quadratic formula:
- x₁ = [5 - √9]/4 = (5 - 3)/4 = 2/4 = 0.5
- x₂ = [5 + √9]/4 = (5 + 3)/4 = 8/4 = 2
- The smallest root is 0.5
You can verify this result using our calculator by entering a=2, b=-5, and c=2.
FAQ
What if the discriminant is negative?
If the discriminant (b² - 4ac) is negative, the quadratic equation has no real roots. The roots will be complex numbers.
Can the smallest root be negative?
Yes, the smallest root can be negative if the quadratic equation has one positive and one negative root. The smallest root will be the negative one.
What if a=0?
If a=0, the equation is no longer quadratic and becomes linear (bx + c = 0). The smallest root calculator is not applicable in this case.
How accurate is the calculator?
The calculator uses standard floating-point arithmetic and should provide accurate results for most practical purposes. For very large or very small numbers, rounding errors may occur.