Problem 11 - Elementary and Intermediate Algebra

Question

Find: \((\sqrt{x}-3)^{2}\)

Step-by-Step Solution

Answer

The expanded form is \(x - 6\sqrt{x} + 9\).

Step 1: Understand the Expression

We are given the expression \((\sqrt{x}-3)^{2}\). This expression is a perfect square trinomial.

Step 2: Apply the Formula for Squaring a Binomial

Recall the formula for squaring a binomial: \( (a - b)^2 = a^2 - 2ab + b^2 \).In our expression, \(a = \sqrt{x}\) and \(b = 3\).

Step 3: Compute Each Term of the Expansion

1. Calculate \(a^2\): \( (\sqrt{x})^2 = x \).2. Calculate \(-2ab\): \(-2 \cdot \sqrt{x} \cdot 3 = -6\sqrt{x} \).3. Calculate \(b^2\): \(3^2 = 9 \).

Step 4: Write the Expanded Expression

Combine the terms from Step 3 to get the expanded form:\(x - 6\sqrt{x} + 9\).

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is a special type of polynomial that is formed by squaring a binomial. It generally has the form \(a^2 \pm 2ab + b^2\). When you expand a perfect square trinomial, you'll notice that the middle term is always twice the product of the two different terms found in the original binomial. This pattern can help us identify and work with perfect square trinomials easily. For example, when given \((\sqrt{x} - 3)^2\), we recognize it as a perfect square trinomial because it fits the pattern mentioned above.

• The first term \(a^2\) is the square of the first term of the binomial.
• The middle term \(\pm 2ab\) is twice the product of the terms in the binomial.
• The last term \(b^2\) is the square of the last term in the binomial.

Knowing how to identify and expand perfect square trinomials is useful for simplifying expressions and solving algebraic problems. Recognizing these patterns helps us efficiently solve algebra exercises and verify our solutions.

The Basics of Binomial Expansion

Binomial expansion is a method used to express the powers of a binomial as a sum of terms. It's a fundamental concept in algebra that lets us simplify complex expressions easily. Particularly, when we talk about squaring a binomial, we're using a specific instance of binomial expansion.For a binomial \((a - b)\), squaring it means we're calculating \((a - b)^2\). This can be expanded to \(a^2 - 2ab + b^2\), involving each term of the binomial in various combinations and powers. Understanding this formula helps when you encounter an expression like \((\sqrt{x} - 3)^2\). Knowing the general rule:

• Begin by squaring the first term directly.
• Then, calculate twice the product of the two terms.
• Finally, square the second term individually.

This step-by-step process simplifies each part of the binomial expansion, ensuring that the solution is neat and comprehensible.

Squaring a Binomial

Squaring a binomial means multiplying the binomial by itself. This is a common operation in algebra, and understanding it is crucial when dealing with quadratic expressions. The process involves using a formula known as the binomial square formula.To square a binomial like \((\sqrt{x} - 3)\), the routine is as follows:1. **Square the first term**: The first term of the binomial \(\sqrt{x}\) is squared to give \(x\).2. **Calculate twice the product of the terms**: Multiply both terms together \(\sqrt{x} \times 3\) and then multiply the result by 2 to deliver \(-6\sqrt{x}\).3. **Square the second term**: The second term \(-3\) when squared results in \(9\).Combining these computations, you obtain the overall expansion, which is \(x - 6\sqrt{x} + 9\). By following these steps and understanding each part of the process, solving similar problems becomes straightforward and manageable.