Solving Quadratic Equations with the Quadratic Formula: A Comprehensive Guide

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The quadratic formula is a mathematical formula used to solve quadratic equations. A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.

The quadratic formula is derived from completing the square method and provides a straightforward way to find the roots (also known as solutions) of any quadratic equation. The formula is as follows:

x = (-b ± √(b^2 – 4ac)) / (2a)

In this formula:
– “±” represents plus or minus, which means that there are two possible solutions for x.
– “√” represents the square root symbol.
– “b^2” represents the square of the coefficient b.
– “4ac” represents the product of the coefficients a and c.

To use the quadratic formula, you simply substitute the values of a, b, and c from your quadratic equation into the formula and perform the calculations. The resulting values for x are the solutions to the equation.

It’s important to note that the quadratic formula will always provide valid solutions for any quadratic equation, regardless of whether the equation has real or complex roots. When the discriminant (the value inside the square root in the formula) is positive, the equation has two distinct real roots. When the discriminant is zero, the equation has a single real root (also known as a double root). And when the discriminant is negative, the equation has two complex conjugate roots.

In summary, the quadratic formula is a powerful tool for solving quadratic equations, providing a direct way to find their roots. It is widely used in various fields of mathematics, physics, and engineering.

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