Problem 12 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression \\( (\sqrt{5x+2} - 4)^2 \\) simplifies to \\( 5x + 18 - 8\sqrt{5x+2} \\).

Step 1: Identify the Expression

Recognize that the expression given is \(\left(\sqrt{5x+2} - 4\right)^2\). This is a power problem involving a binomial.

Step 2: Apply the Binomial Square Formula

The formula for squaring a binomial \( (a-b)^2 \) is \( a^2 - 2ab + b^2 \). Identify \( a = \sqrt{5x+2} \) and \( b = 4 \) in the expression.

Step 3: Calculate the Square of the First Term

Square the term \( a = \sqrt{5x+2} \) to get \( (\sqrt{5x+2})^2 = 5x+2 \).

Step 4: Calculate the Middle Term

Calculate \(-2ab\), which is \(-2(\sqrt{5x+2})(4) = -8\sqrt{5x+2}\).

Step 5: Calculate the Square of the Second Term

Square \( b = 4 \) to get \( b^2 = 16 \).

Step 6: Formulate the Expanded Expression

Combine all the parts: \( 5x+2 - 8\sqrt{5x+2} + 16 \).

Step 7: Simplify the Expression

Combine like terms: \( 5x + 18 - 8\sqrt{5x+2} \). This yields the final expression.

Key Concepts

Square of a Binomial

The square of a binomial is a common operation in algebra that simplifies expressions. A binomial is simply an algebraic expression with two terms, like

• \( (a + b) \)
• \( (a - b) \)

To square a binomial, you use the formula:\[(a \, \pm \, b)^2 = a^2 \, \pm \, 2ab \, + \, b^2\] Breaking this down:

• \(a^2\) represents the square of the first term.
• \(2ab\) adds the product of the two terms, multiplied by 2.
• \(b^2\) is the square of the second term.

In our exercise, the expression is \((\sqrt{5x+2} - 4)^2\). Identifying our terms, we have:

• \(a = \sqrt{5x+2}\)
• \(b = 4\)

Applying the formula step by step, we start by squaring each term followed by calculating the middle term. Finally, it all combines into a clean expression. This expansion method helps in simplifying and solving equations efficiently.

Algebraic Expressions

Algebraic expressions are the heart of algebra. They consist of variables, numbers, and operation symbols. There are no equal signs, which differentiates them from equations. Examples might be:

• \( 3x + 7 \)
• \( x^2 - 5x + 6 \)

In our initial expression \((\sqrt{5x+2} - 4)^2\), the important part is recognizing each component of the binomial. Here, \(\sqrt{5x+2}\) is a radical expression involving a linear term inside a square root. Along with "-4", these components form an expression that needs to be expanded.When working with algebraic expressions, remember:

• Identify variables and constants.
• Use the rules of algebra, like the distributive property.
• Simplify step-by-step by combining like terms.

This organized approach helps in managing more complex operations like the one in our exercise.

Radical Expressions

Radical expressions contain a radical sign (√), such as square roots. They're quite common in various math problems and appear in algebra frequently. For example:

• \( \sqrt{9} \) which simplifies to 3
• \( \sqrt{x^2} \) which simplifies to \(x\) if \(x\) is non-negative.

In the provided expression \((\sqrt{5x+2} - 4)^2\), the radical part \(\sqrt{5x+2}\) simply stays as is in parts of the solution.
Squaring the radical \(\sqrt{5x+2}\) in the expansion results in eliminating the square root, simplifying this part to \(5x + 2\).

When handling radical expressions:

• Simplify the radical if possible before combining with other terms.
• Work carefully to avoid mistakes when it involves both radicals and other numbers.
• Always check if the entire expression under the radical can be simplified too.

Understanding radicals ensures accuracy and simplifies solving for solutions in algebra.