91Ó°ÊÓ

A binomial is an algebraic expression with two terms, like \((a + b)\) or \((a - b)\). Binomial expansion involves multiplying out an expression that is raised to a power, such as \((a + b)^2\) or \((a - b)^2\). Here, you used the formula for binomial expansion to simplify \( (3 - \sqrt{2})^2 \). Let's break down the steps.
First, remember the expansion formula: \((a - b)^2 = a^2 - 2ab + b^2\).

In this problem:
\((3 - \sqrt{2})^2\) means:
\(a = 3\) and
\(b = \sqrt{2}\)

Plug them into the expansion formula:
\(3^2 - 2(3)(\sqrt{2}) + (\sqrt{2})^2 \)

By systematically applying this expansion, binomials can be simplified easily.
Now, the next step involves...

Radical simplification is the process of simplifying expressions involving roots, such as square roots or cube roots. In the previous example, you had to simplify terms involving \( \sqrt{2} \). Let's see how it works.
You expanded the binomial to get:
\((3 - \sqrt{2})^2 = 3^2 - 2(3)(\sqrt{2}) + (\sqrt{2})^2 \).

Now, compute each term one by one:
\( - 3^2 = 9 \)
\( - 2(3)(\sqrt{2}) = 6\sqrt{2} \)
\( - (\sqrt{2})^2 = 2 \)

Substitute back the computed values into the expansion, you get:
\(9 - 6\sqrt{2} + 2\).
Finally, combine the constants 9 and 2:
It's \(9 + 2 = 11 - 6\sqrt{2}\).

Simplifying radicals is crucial to dealing with expressions containing square roots effectively.

Quadratic functions are algebraic expressions where the highest exponent of the variable is 2. In this exercise, you started with \(f(x)=x^2 \).
A quadratic function is typically written as \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants. Here, \(a=1,b=0, \) and \(c=0\).

These functions graph as parabolas.
When you evaluated a quadratic function by substituting a value for \(x\), you essentially determined a specific point on this parabola.
In this exercise, you substituted \(x = 3 - \sqrt{2}\) into\ \(f(x)=x^2 \), this is evaluated to find \(f(3-\root{2})=\text{(3\root{2)}^2 \).
\ In short, evaluating quadratic functions is a key skill in algebra as it allows for understanding the behavior and shape of parabolas in different contexts. <\br>