Problem 11 - 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 Problem

We need to expand the expression \((\sqrt{x}-3)^{2}\). This means we must express it in a form without the power notation.

Step 2: Apply the Binomial Theorem

Use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{x}\) and \(b = 3\).

Step 3: Calculate \(a^2\)

Compute \((\sqrt{x})^2\). Since squaring a square root returns the original expression under the root, \((\sqrt{x})^2 = x\).

Step 4: Calculate \(-2ab\)

Compute \(-2 \times \sqrt{x} \times 3 = -6\sqrt{x}\).

Step 5: Calculate \(b^2\)

Compute \(3^2\), which is \(9\).

Step 6: Combine Terms

Combine the results of Steps 3, 4, and 5: \(x - 6\sqrt{x} + 9\). This is the expanded form of \((\sqrt{x} - 3)^2\).

Key Concepts

Square Root Properties

Square roots are an essential concept in algebra. They represent a value that, when multiplied by itself, gives the original number under the root. For instance, the square root of 9 is 3, because 3 multiplied by itself equals 9. Understanding square roots can help simplify algebraic expressions involving roots.

One fundamental property of square roots is that the square of a square root returns the original number. Mathematically, \((\sqrt{a})^2 = a\). This property is crucial when expanding binomial expressions with roots. As seen in our exercise, when you square \(\sqrt{x}\), you simplify to just \(x\). Knowing this property aids in accurately simplifying and expanding expressions.

There are also important rules when combining square roots, such as:

• \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
• \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

These rules help in manipulating and simplifying expressions with multiple roots involved, making the calculations easier to follow and solve.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They usually appear in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our exercise, we worked with a special type—a binomial squared. Understanding how to handle quadratic expressions is essential for solving, graphing, and analyzing algebraic equations.

When you encounter a squared binomial like \((a - b)^2\), you use the binomial theorem to expand it into a quadratic expression. The resulting expression is \(a^2 - 2ab + b^2\). This expansion helps to visualize the equation better and solve it effectively.

Completing the square and factoring are typical methods used in dealing with quadratic expressions. They help find the roots of the equations or convert them into vertex form for graphing purposes. Mastery of these methods provides a robust toolkit for handling various algebraic problems.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving algebraic expressions and equations. It involves rearranging and simplifying expressions to make them easier to work with. The goal is often to isolate a variable or to break down complex expressions into simpler parts.

Some of the key techniques include:

• Distributing or expanding expressions like \((a + b)(a - b)\)
• Combining like terms to condense expressions
• Applying properties of operations, such as the commutative and associative properties

In our exercise, algebraic manipulation involved using the binomial formula to expand \((\sqrt{x} - 3)^2\) into a simpler form. We expanded and simplified the expression by computing each part step by step. This includes expanding \((\sqrt{x})^2\) to \(x\), calculating \(-2 \times \sqrt{x} \times 3\) to \(-6\sqrt{x}\), and computing the constant part.

Effective algebraic manipulation is all about precision and following logical steps to simplify expressions or solve equations efficiently. The more you practice, the more intuitive these techniques become, allowing for smoother problem-solving.