Problem 61 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{4 x^{2}+32 x+64}+\sqrt{10 x^{2}+80 x+160} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(\sqrt{4x^2+32x+64}+\sqrt{10x^2+80x+160}\). Answer: \((x+4)\left[2 + \sqrt{10(x+4)^2}\right]\).

Step 1: Find perfect square in the first expression under the root

We first factorize the first expression under the root to find a perfect square. The expression is, \(4x^2 + 32x + 64\). Factoring, we have: \(4(x^2+8x+16)\). Observe that \((x^2+8x+16)\) is a perfect square of \((x+4)^2\). So, our expression becomes: \(\sqrt{4(x+4)^2}\).

Step 2: Take the square root of the first expression

Since \(\sqrt{4(x+4)^2}\) is a product of squares, we can simplify it as: $$\sqrt{4}\cdot\sqrt{(x+4)^2} = 2(x+4).$$

Step 3: Factorize the second expression under the root

Now, we will factorize the second expression to simplify it further. The second expression is \(10x^2 + 80x + 160\). Factoring, we get: $$10(x^2+8x+16).$$ Observe that \((x^2+8x+16)\) is also a perfect square, which is \((x+4)^2\). So our expression becomes: $$\sqrt{10(x+4)^2}.$$

Step 4: Combine both expressions

Now, we have simplified both expressions and can combine them as follows: $$2(x+4) + \sqrt{10(x+4)^2}.$$ As both terms have \((x+4)\) as a common factor, the final simplified expression is: $$(x+4)\left[2 + \sqrt{10(x+4)^2}\right].$$

Key Concepts

Perfect Square

A perfect square is a number or expression that can be represented as the square of another whole number or expression. For example, the expression - \((x+4)^2\) is a perfect square because it is obtained by multiplying the expression - \((x+4)\) by itself. When you encounter a quadratic expression like - \(x^2 + 8x + 16\) within an algebraic problem, it is important to identify it as a perfect square to simplify operations like finding square roots. Recognizing a perfect square allows you to rewrite the expression in a more workable form, such as: - \((x+4)^2\), which makes it easier to compute further operations like roots.

Factoring

Factoring involves breaking down an expression into a product of simpler expressions called factors. This method is crucial for simplifying algebraic expressions and solving equations. For example, consider the expression - \(4x^2 + 32x + 64\). By factoring out the common factor, we express it as: - \(4(x^2 + 8x + 16)\). By recognizing - \(x^2 + 8x + 16\) as a perfect square, we can further rewrite it to - \((x+4)^2\). This simplifies the expression under the root and allows for easier mathematical manipulation. Factoring is a key skill that helps in finding common factors and simplifying expressions so that they become easier to manage or solve.

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, results in the original number. Taking the square root of a perfect square simplifies the expression significantly. In the context of our solution, for the expression - \(4(x+4)^2\), we can take the square root to break it down, which results in: - \(\sqrt{4} \cdot \sqrt{(x+4)^2}\). This simplifies further to - \(2(x+4)\). Understanding how to take the square root of perfect squares can be incredibly helpful in reducing the complexity of algebraic expressions. It is a fundamental concept used frequently in algebra and other areas of mathematics.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). Simplifying algebraic expressions means reducing them to their simplest form, which often involves factoring and applying arithmetic operations. The given problem initially provides complex expressions like - \(\sqrt{4x^2+32x+64} + \sqrt{10x^2+80x+160}\). To simplify these, we break down each expression into more manageable parts using techniques like factoring and recognizing perfect squares. Once each part is simplified, we can combine them. Simplifying algebraic expressions makes them easier to work with and solve further mathematical problems. Through this process, complex-looking expressions can yield surprising insights and clearer outcomes.