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Quadratic and Square Root Equation Calculator

Reviewed by Calculator Editorial Team

This calculator solves quadratic equations of the form ax² + bx + c = 0 and calculates square roots of positive numbers. It provides both exact and approximate solutions, along with visual representations of the results.

How to Use This Calculator

To solve a quadratic equation:

  1. Enter the coefficients a, b, and c in the quadratic equation section
  2. Select whether you want exact solutions or decimal approximations
  3. Click "Calculate Quadratic Equation"

To calculate a square root:

  1. Enter a positive number in the square root section
  2. Select the precision for the decimal approximation
  3. Click "Calculate Square Root"

The calculator will display the results in the right panel, along with a visual chart showing the relationship between the inputs and outputs.

Quadratic Equations

A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0. The solutions to this equation are called roots. The quadratic formula, which is derived from completing the square, provides the roots of any quadratic equation:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

  • If the discriminant is positive, there are two distinct real roots
  • If the discriminant is zero, there is exactly one real root (a repeated root)
  • If the discriminant is negative, there are two complex conjugate roots

Square Root Calculations

The square root of a number x is a value y such that y² = x. For positive real numbers, there are two square roots: one positive and one negative. The principal (or non-negative) square root is denoted by √x.

Square roots can be calculated using various methods:

  1. Exact form: √x when x is a perfect square
  2. Decimal approximation: using iterative methods like Newton's method
  3. Logarithmic method: using the identity √x = e^(0.5 * ln(x))

This calculator provides both exact forms (when possible) and decimal approximations with configurable precision.

Worked Examples

Example 1: Quadratic Equation

Solve x² - 5x + 6 = 0

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 = [5 ± √1] / 2

This gives two solutions: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2

Example 2: Square Root Calculation

Calculate √25

The exact form is 5, since 5² = 25. The decimal approximation is also 5.000000.

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

The discriminant is negative (4 - 20 = -16), so the solutions are complex:

x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2 = -1 ± 2i

Frequently Asked Questions

What is the difference between a quadratic equation and a square root calculation?
A quadratic equation is a second-degree polynomial equation that can have up to two solutions. A square root calculation finds a number that, when multiplied by itself, gives the original number. They are related through the quadratic formula, which involves square roots.
How do I know if a quadratic equation has real solutions?
A quadratic equation has real solutions if the discriminant (b² - 4ac) is zero or positive. If the discriminant is negative, the solutions will be complex numbers.
What is the difference between exact and approximate square roots?
Exact square roots are simplified radical forms (like √2) when possible. Approximate square roots are decimal representations (like 1.4142) that can be calculated to any desired precision.
Can this calculator solve equations with variables other than x?
This calculator is designed to solve for x in equations of the form ax² + bx + c = 0. It does not currently support solving for other variables.