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Find The Positive Root Calculator

Reviewed by Calculator Editorial Team

A positive root of a quadratic equation is a solution where x is greater than zero. This calculator helps you find the positive root of equations in the form x² + bx + c = 0 using the quadratic formula.

What is a positive root?

In mathematics, a root of a quadratic equation is a value of x that satisfies the equation. A positive root is specifically one where x > 0. For example, in the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3. The positive root here is 3.

Key Points

  • Quadratic equations have up to two real roots
  • The positive root is the solution greater than zero
  • If both roots are positive, both are considered positive roots
  • If no roots are positive, the equation has no positive roots

Finding positive roots is important in many real-world applications, including physics, engineering, and economics. For example, in projectile motion problems, positive roots often represent valid solutions to the problem.

Quadratic formula

The quadratic formula is the standard method for finding the roots of a quadratic equation. The formula is:

Quadratic Formula

For an equation ax² + bx + c = 0, the roots are given by:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are coefficients from the quadratic equation
  • √(b² - 4ac) is the discriminant
  • The ± symbol indicates there are two possible solutions

To find the positive root, you calculate both solutions and select the one that is greater than zero. If both solutions are positive, both are valid positive roots.

How to use this calculator

  1. Enter the coefficients a, b, and c from your quadratic equation
  2. Click "Calculate" to find the roots
  3. The calculator will display the positive root(s) if they exist
  4. Review the detailed solution and chart visualization

Important Notes

  • This calculator only finds positive roots
  • If the discriminant is negative, there are no real roots
  • If both roots are positive, both will be displayed
  • Results are rounded to 4 decimal places

Worked example

Let's solve x² - 5x + 6 = 0 using this calculator:

  1. Identify coefficients: a = 1, b = -5, c = 6
  2. Calculate discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Apply quadratic formula: x = [5 ± √1]/2
  4. Calculate both roots:
    • x₁ = (5 + 1)/2 = 3
    • x₂ = (5 - 1)/2 = 2
  5. Both roots are positive, so both are valid solutions

Using our calculator with these values would show both 2 and 3 as positive roots.

FAQ

What if the discriminant is negative?

If the discriminant (b² - 4ac) is negative, the equation has no real roots. This means there are no positive roots either, as all roots would be complex numbers.

What if both roots are positive?

If both roots calculated from the quadratic formula are positive, both will be displayed as positive roots. This happens when the parabola intersects the x-axis at two points above zero.

Can I use this calculator for non-standard forms?

This calculator works with standard quadratic equations in the form ax² + bx + c = 0. For other forms, you may need to rewrite the equation first.

How accurate are the results?

Results are calculated using JavaScript's Math functions and are accurate to 15 decimal places internally, then rounded to 4 decimal places for display.