Problem 12 - Intermediate Algebra

Question

Find: \((\sqrt{5 x+2}-4)^{2}\)

Step-by-Step Solution

Answer

The expanded form is \(5x + 18 - 8\sqrt{5x+2}\).

Step 1: Recognize the Square of a Binomial

The given expression is \[(\sqrt{5x+2}-4)^2 \].This is in the form of \((a-b)^2\), where \(a = \sqrt{5x+2}\) and \(b = 4\). To solve, we apply the identity for the square of a binomial: \((a-b)^2 = a^2 - 2ab + b^2\).

Step 2: Calculate the Square of the First Term (\(a^2\))

The first term \(a\) is \(\sqrt{5x+2}\). So, the square of the first term is:\[(\sqrt{5x+2})^2 = 5x + 2.\]

Step 3: Calculate the Product (-2ab)

The product \(-2ab\) is calculated as follows:\[-2 \cdot \sqrt{5x+2} \cdot 4 = -8\sqrt{5x+2}.\]

Step 4: Calculate the Square of the Second Term (\(b^2\))

The second term \(b\) is 4. The square of the second term is:\[4^2 = 16.\]

Step 5: Combine All Terms from Steps 2, 3, and 4

Now we combine the results from the previous steps:\[(\sqrt{5x+2})^2 - 2(\sqrt{5x+2})(4) + 4^2 = 5x + 2 - 8\sqrt{5x+2} + 16.\] This simplifies to:\[5x + 18 - 8\sqrt{5x+2}.\]

Key Concepts

Square of a Binomial

One of the fundamental concepts in Intermediate Algebra is the "Square of a Binomial," which refers to the expansion of an expression in the form \((a-b)^2\). A binomial is simply an expression with two terms, such as \(a - b\) in our example. When squaring a binomial, you can use the identity: \((a-b)^2 = a^2 - 2ab + b^2\). This identity lets us expand a binomial squared to a form that eliminates parentheses, making calculations easier.

• "\(a^2\)" denotes the square of the first term in the binomial.
• "\(-2ab\)" indicates the product of the first term \( a\) and the second term \( b\), multiplied by 2, ensuring any middle term is accounted for.
• "\(b^2\)" represents the square of the second term.

Applying this identity approves an organized approach to tackle expressions like \((\sqrt{5x+2}-4)^2\). Step by step, the expanded form allows us to analyze each part independently before combining them again.

Algebraic Expressions

Algebraic expressions form the foundation of algebra, involving numbers, variables, and operation signs. In our exercise, the expression \((\sqrt{5x+2}) - 4\) is an example of such an expression. Breaking down the algebraic expression helps us understand the components:

• The term \(\sqrt{5x+2}\) comprises a radical expression and a simple algebraic expression inside it.
• The constant \(-4\) is a straightforward number, which can combine with other constants.

Algebraic expressions can describe not just simple operations, but also complex combinations of terms and functions. Learning how to manipulate these by expanding, factoring, or simplifying makes solving equations more efficient. Understanding the behavior of each part of an algebraic expression can empower you to handle equations in logical steps.

Radical Expressions

Radical expressions include the square root, cube root, or other root symbols. In our expression, \(\sqrt{5x+2}\) is a radical expression and plays a key role in our problem. The radical symbol \(\sqrt{}\) represents the square root of what lies within it. Key points about radical expressions include:

• They cannot always be simplified to integers or even rational numbers without approximation.
• They are subject to the rules of exponents, such as \((\sqrt{x})^2 = x\). This is visible in our exercise where squaring \(\sqrt{5x+2}\) efficiently removes the radical.

Handling radical expressions requires special attention to ensure they are managed correctly, especially when combined with other terms.It’s vital to apply correct algebraic rules when incorporating radicals into larger expressions to maintain accuracy in the result. Getting comfortable with radical expressions unlocks more complex algebraic and geometric problems.