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Squaring binomials is an important concept in algebra. Binomials are algebraic expressions that contain two terms, like \( a + b \) or \( t - 11 \). When you square a binomial, you are multiplying it by itself. The general formula for squaring a binomial is: \[ (a \pm b)^2 = a^2 \pm 2ab + b^2 \].

Notice how we have three terms in the result. Here's a quick breakdown of each part:

• The first term, \( a^2 \), is the square of the first term in the binomial.
• The second term, \( 2ab \), is twice the product of the two terms in the binomial.
• The third term, \( b^2 \), is the square of the second term in the binomial.

This formula works for both positive and negative binomials.
For example, for \( (t - 11)^2 \), \( a = t \) and \( b = 11 \), so we apply the formula to get the expanded form: \[ t^2 - 2(t)(11) + 11^2 \].
Then, we simplify it further to obtain: \[ t^2 - 22t + 121 \].

Algebraic expressions are combinations of numbers, variables, and operations. In the expression \( t - 11 \), \ 't' \ is a variable, and \ -11 \ is a constant term.
When dealing with algebraic expressions:

• Variables represent unknown values and can change.
• Constants are fixed values.
• Operations like addition, subtraction, multiplication, and division combine these elements.

Using algebraic expressions, we can model real-world scenarios, solve equations, and much more. For example, squaring the binomial \( t - 11 \) results in the more complex expression \ t^2 - 22t + 121 \. This expanded expression still represents the same value as \( (t - 11)^2 \), but it shows the relationship between terms more explicitly.

Simplifying expressions makes them easier to work with and understand. It involves combining like terms and performing arithmetic operations.
In the case of the expression \( (t - 11)^2 \), we first expanded it to get \ t^2 - 22t + 121 \. Now, we simplify it by:

• Calculating any numerical coefficients.
• Combining like terms.

Let's break down each step:

1. In the term \( -2(t)(11) \), we multiply the coefficients: \ -2 \times 11 = -22 \.
2. So, the middle term becomes \ -22t \.
3. Adding \ 11^2 \ gives us \ 121 \.
Now the expression is simplified to \ t^2 - 22t + 121 \. Remember, simplification does not change the value of the expression or equation; it just makes it clearer and easier to manage.
By mastering the art of simplifying expressions, you can solve complex problems systematically and accurately.