Quadratic Equation Calculator

Quadratic Equation Calculator User Manual

1. Introduction

This document explains how to use the Quadratic Equation Calculator and provides a mathematical background on quadratic equations. The calculator computes the solutions to a quadratic equation ((ax^2 + bx + c = 0)) based on the user-entered coefficients (a), (b), and (c).

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2. How to Use the Calculator

Getting Started

To use the calculator, you need to input the values of the coefficients (a), (b), and (c):

1. Coefficient (a):

• The coefficient of the quadratic term ((x^2)).
• (a) must not be zero.

1. Coefficient (b):

• The coefficient of the linear term ((x)).

1. Coefficient (c):

• The constant term.

Enter the values in the respective fields. The calculator will automatically compute the solutions as you modify the coefficients.

Interpreting the Results

The calculator provides solutions based on the discriminant ((\Delta = b^2 – 4ac)):

• Complex Roots:
• If the discriminant is negative, the solutions include real and imaginary parts.
• Double Root:
• If the discriminant is zero, a single real root (repeated root) is returned.
• Two Distinct Real Roots:
• If the discriminant is positive, two different real roots are returned.

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3. Mathematical Background of Quadratic Equations

Quadratic Equations and the Discriminant

A quadratic equation is generally expressed in the form:

[
ax^2 + bx + c = 0
]

Where:

• (a), (b), and (c) are real coefficients.
• (a \neq 0).

The discriminant ((\Delta)) is calculated as:

[
\Delta = b^2 – 4ac
]

The discriminant determines the nature of the roots:

• (\Delta > 0): Two distinct real roots.
• (\Delta = 0): One real root (double root).
• (\Delta < 0): Two complex roots.

Relation Between Roots and Coefficients

Let the two roots of the quadratic equation be (x_1) and (x_2). The following relationships hold:

1. Sum of Roots:
   [
   x_1 + x_2 = -\frac{b}{a}
   ]
2. Product of Roots:
   [
   x_1 \times x_2 = \frac{c}{a}
   ]

These properties help in understanding the relationship between the roots and the coefficients of the quadratic equation.

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This manual provides a comprehensive guide to using the Quadratic Equation Calculator and understanding the underlying mathematical concepts. Quadratic equations have wide applications in fields like physics, engineering, and business, making this tool highly practical for solving real-world problems.