Problem 75 - Algebra 2

Question

Find each power. $$ (2 \sqrt{x}-3)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(4x - 12 \sqrt{x} + 9\).

Step 1: Identify the Expression

The given expression is \((2 \sqrt{x} - 3)^2\). This is a binomial expression raised to the power of 2.

Step 2: Recall the Binomial Formula

To solve \((a - b)^2\), use the formula \(a^2 - 2ab + b^2\). This simplifies the expansion of the square of a binomial.

Step 3: Expand the Binomial

Apply the formula: \((2 \sqrt{x})^2 - 2(2 \sqrt{x})(3) + 3^2\).

Step 4: Calculate Each Term

1. Calculate \((2 \sqrt{x})^2\): \((2 \sqrt{x})^2 = 4x\).2. Calculate \(-2(2 \sqrt{x})(3)\): \(-2 \times 2 \sqrt{x} \times 3 = -12 \sqrt{x}\).3. Calculate \(3^2\): \(3^2 = 9\).

Step 5: Combine Terms

Combine the calculated terms from Step 4: \(4x - 12 \sqrt{x} + 9\).

Step 6: Finalize Solution

The expanded and simplified form of \((2 \sqrt{x} - 3)^2\) is \(4x - 12 \sqrt{x} + 9\).

Key Concepts

Square of a Binomial

The square of a binomial involves multiplying a binomial by itself. A binomial is an algebraic expression composed of two distinct terms, such as

• \((a + b)\) or \((a - b)\).

When you square a binomial, it means you take the expression and multiply it by itself. The general formula for squaring a binomial \((a + b)^2\) is:

• \(a^2 + 2ab + b^2\)

For another similar expression \((a - b)^2\), the formula is:

• \(a^2 - 2ab + b^2\)

These formulas are crucial as they simplify the process of binomial expansion.
In the context of the exercise, squaring the expression

• \((2 \sqrt{x} - 3)\),

you apply the formula \(a^2 - 2ab + b^2\) to easily expand the expression.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators such as plus, minus, multiplication, and division signs. These expressions can consist of one or several terms, where each term can be a number (constant), variable (unknown value represented by letters), or a product of numbers and variables.
In the exercise, the algebraic expression given is

• \((2 \sqrt{x} - 3)^2\)

which includes a combination of numbers and a variable \(x\), making it a binomial.
When working with these expressions, it is important to identify the different parts:

• \(2 \sqrt{x}\) - a term with a variable.
• \(-3\) - a constant term.

These components help in accurately applying algebraic rules and formulas, like the binomial square formula mentioned earlier, which is vital for correctly solving the expression given.

Simplifying Expressions

Simplifying expressions means breaking down complex algebraic expressions into simpler, more manageable forms without changing their value. The purpose is to make calculations easier and solutions clearer.
In the exercise, once you expand the square of a binomial

• \((2 \sqrt{x} - 3)^2\),

the next step is to simplify it:

• Calculate each term like \((2 \sqrt{x})^2\) which equals \(4x\),
• \(-2(2 \sqrt{x})(3)\) which equals \(-12 \sqrt{x}\),
• and \(3^2\) which equals \(9\).

Finally, you combine these values to get the simplified expression:

• \(4x - 12 \sqrt{x} + 9\).

Simplification not only makes expressions easier to understand but also helps in identifying equivalent expressions and solving equations more efficiently.