91Ó°ÊÓ

A quadratic equation is typically written in the form: \[ ax^2 + bx + c = 0 \]
In this equation, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.
The roots (or solutions) of a quadratic equation are the values of \(x\) that satisfy the equation.

For example, if the roots given are \(2 - \sqrt{10}\) and \(2 + \sqrt{10}\), these can be plugged back into the quadratic equation to verify they are indeed the solutions.
The quadratic equation that has these roots can be found using the relationships:

• Sum of the roots = \(\alpha + \beta = -\frac{b}{a}\)
• Product of the roots = \(\alpha \beta = \frac{c}{a}\)

In this case, the equation forms as follows:

Step 1: Identify Roots:
The roots \(2 - \sqrt{10}\) and \(2 + \sqrt{10}\)
Step 2: Form Factorized Equation:
Use the roots to form the factorized version: \[ (x - (2 - \sqrt{10})) (x - (2 + \sqrt{10})) \]

The difference of squares formula is a useful algebraic identity. It states:

\[ (a - b)(a + b) = a^2 - b^2 \]

This formula can be applied to a variety of algebraic problems, especially in factoring and simplifying expressions.

In our exercise, we use the roots \(2 - \sqrt{10}\) and \(2 + \sqrt{10}\) to create the factorized form.

Applying the difference of squares formula, we get:

\[ (x - 2 + \sqrt{10})(x - 2 - \sqrt{10}) = ((x - 2) - \sqrt{10})((x - 2) + \sqrt{10}) \]

By applying the formula:

• let \(a = (x-2)\)
• let \(b = \sqrt{10}\)

Therefore:

\[ a^2 - b^2 = (x-2)^2 - (\sqrt{10})^2 \]

Which simplifies to:

\[ (x - 2)^2 - 10 \]

Factoring refers to breaking down an equation or a number into its multiplicative components. The goal is to rewrite the quadratic equation in a simpler, multiplied form.

Beginning with the quadratic expression:

\[ ((x - 2) - \sqrt{10})((x - 2) + \sqrt{10}) \]

First, use the difference of squares formula to obtain:

\[ ((x - 2)^2 - 10) \]

Next, we expand \((x - 2)^2\):

\[ (x - 2)(x - 2) = x^2 - 4x + 4 \]

Putting it all together:
\[ x^2 - 4x + 4 - 10 \]

Simplifying further, we combine like terms to get:

\[ x^2 - 4x - 6 \]

Thus, the final quadratic equation with roots \(2 \pm \sqrt{10}\) is:

\[ x^2 - 4x - 6 = 0 \]