The Quadratic Formula Explained: Understanding the Roots and Formulation

Quadratic equations are common in many areas of mathematics and science. The standard form of a quadratic equation is given by:

General Form of a Quadratic Equation

The general form of a quadratic equation is:

ax2 bx c 0

Where:

a is the coefficient of the x2 term b is the coefficient of the x term c is the constant term

The roots of this equation (the values of x that satisfy the equation) can be found using the quadratic formula, which is an important tool in algebra. The quadratic formula is:

The Quadratic Formula

x u00BD(-b u00B1 u221A(b2 - 4ac))

This formula appears in the general quadratic equation ax2 bx c 0. The plus-minus symbol (u00B1) indicates that there are two roots for the equation:

The first root is ( frac{-b - sqrt{b^2 - 4ac}}{2a} ) The second root is ( frac{-b sqrt{b^2 - 4ac}}{2a} )

The quadratic formula is derived from the standard form of the quadratic equation. Let's delve into how it is derived and its usage.

Deriving the Quadratic Formula

Starting from the general form of a quadratic equation:

ax2 bx c 0

We can determine the sum and product of the roots:

Sum and Product of the Roots

The sum of the roots ( alpha beta ) and the product of the roots ( alpha beta ) are given by:

Sum of the roots: ( alpha beta -frac{b}{a} ) Product of the roots: ( alpha beta frac{c}{a} )

Using the sum and product of the roots, we can derive the difference of the roots:

Deriving the Difference of the Roots

Let's start by squaring the sum of the roots:

(alpha beta)2 alpha2 2alpha*beta beta2

Given that ( alpha beta -frac{b}{a} ) and ( alpha beta frac{c}{a} ), we have:

(-frac{b}{a})2 alpha2 2*alpha*beta beta2

Rearranging terms and solving for the difference of the roots ( alpha - beta ), we get:

alpha - beta u221A((-frac{b}{a})2 - 4*(frac{c}{a}))

Simplifying further:

alpha - beta u221A(frac{b2 - 4ac}{a2})

alpha - beta u221A(frac{b2 - 4ac}{acdot a})

alpha - beta (frac{sqrt{b2 - 4ac}}{a})

Using the Sum and Difference to Find the Roots

With the sum and difference of the roots, we can use the quadratic formula to find the roots:

x (frac{-b pm sqrt{b2 - 4ac}}{2a})

This formula can be understood as:

x (frac{-b - sqrt{b2 - 4ac}}{2a})

and

x (frac{-b sqrt{b2 - 4ac}}{2a})

For example, if we have the quadratic equation:

2x2 5x - 3 0

Here, a 2, b 5, and c -3. Plugging these values into the quadratic formula:

x (frac{-5 pm sqrt{52 - 4 cdot 2 cdot (-3)}}{2 cdot 2})

x (frac{-5 pm sqrt{25 24}}{4})

x (frac{-5 pm sqrt{49}}{4})

x (frac{-5 pm 7}{4})

Solving for the two roots:

x (-frac{5 7}{4} -3) x (-frac{5 - 7}{4} (frac{1}{2})

Therefore, the roots of the equation 2x2 5x - 3 0 are -3 and (frac{1}{2}).

Understanding the quadratic formula and its derivation is crucial for solving quadratic equations in a variety of applications. Whether in physics, engineering, or mathematics itself, the quadratic formula is a powerful tool to have in your toolkit.