Question

Find the product.\((5+\sqrt{x})(5-\sqrt{x})\)

Short answer

The product \((5+\sqrt{x})(5-\sqrt{x})\) equals \(25 - x\)

Step-by-step solution

Identify the terms in the given binomials

In the given expression \((5+\sqrt{x})(5-\sqrt{x})\), \(a\) is identified as 5, and \(b\) is identified as \(\sqrt{x}\)

Apply the difference of squares formula

Applying the formula, the product can be written as \( a^2 - b^2 \). In place of \(a\) and \(b\), put 5 and \(\sqrt{x}\) respectively. So, the expression becomes \(5^2 - (\sqrt{x})^2\)

Calculate the result

By calculating we get \(25 - x\), which is the required product.

Key concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables (like x or y), and operation symbols such as plus (+), minus (-), multiplication (×), and division (÷). They represent quantities without fixed values, often unknowns or variables. Algebraic expressions are essential because they can depict real-life situations in an abstract way, allowing us to solve a variety of problems.

For instance, the expression \(5+\sqrt{x}\) is an algebraic expression. Here, 5 is a constant, and \(\sqrt{x}\) represents a variable within a radical. The beauty of algebraic expressions lies in their ability to model relationships and changes in quantities.

Binomial Products

The term 'binomial' refers to an algebraic expression containing two terms, such as \(a+b\) or \(a-b\). When two binomials are multiplied, they form what's known as a binomial product. The process of multiplying binomials can often lead to specific patterns in algebra. One special pattern is known as the difference of squares.

In the case of our example \(5+\sqrt{x}\) and \(5-\sqrt{x}\), we can consider 5 and \(\sqrt{x}\) as \(a\) and \(b\) respectively. When multiplied, the result is \(a^2 - b^2\), which is a direct application of the difference of squares formula. This pattern is useful because it simplifies the multiplication of two binomials into a single, more manageable expression.

Radicals in Algebra

Radicals in algebra involve the use of the radical sign \(\sqrt{~}\) to denote the square root or higher roots of an expression. The term under the square root is called the radicand. Working with radicals is quite common in algebra, especially in simplifying expressions and solving equations.

Radicands may contain both numbers and variables; for example in \(\sqrt{x}\), x is expected to be a non-negative number if we are dealing with real numbers only. The key is to understand how to simplify radicals by either factorizing the radicand or rationalizing the denominator when needed. In our exercise example, the radical \(\sqrt{x}\) is squared in the binomial product, resulting in the radicand itself, which is x.