Problem 53 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$(2 \sqrt{x}-5)(3 \sqrt{x}+1)$$

Step-by-Step Solution

Answer

\(6x - 13\sqrt{x} - 5\)

Step 1: Apply the distributive property

To multiply two binomials, use the distributive property (also known as the FOIL method for binomials). First, multiply each term in the first binomial by each term in the second.\((2\sqrt{x})(3\sqrt{x}) + (2\sqrt{x})(1) + (-5)(3\sqrt{x}) + (-5)(1)\)

Step 2: Simplify each term

Simplify each of the products obtained in the previous step. 1. \((2\sqrt{x})(3\sqrt{x}) = 6x\) 2. \((2\sqrt{x})(1) = 2\sqrt{x}\) 3. \((-5)(3\sqrt{x}) = -15\sqrt{x}\) 4. \((-5)(1) = -5\)

Step 3: Combine like terms

Add all the simplified terms together from Step 2. Combine the terms that are similar, in this case, the terms involving \(\sqrt{x}\).\(6x + 2\sqrt{x} - 15\sqrt{x} - 5\)

Step 4: Simplify the expression

Combine the like terms involving \(\sqrt{x}\) to further simplify the expression.\(6x + (2 - 15)\sqrt{x} - 5 = 6x - 13\sqrt{x} - 5\)

Step 5: Final answer

The simplified expression after combining like terms is:\(6x - 13\sqrt{x} - 5\)

Key Concepts

Distributive Property

When multiplying binomials like \((2 \sqrt{x} - 5)(3 \sqrt{x} + 1)\), we often use a fundamental mathematical rule called the distributive property. This property lets us break down complex operations into simpler ones. It states that each term in the first binomial should be multiplied by each term in the second binomial. In this example, we distribute the terms from the first binomial:

• \(2\sqrt{x} \times 3\sqrt{x} = 6x\)
• \(2\sqrt{x} \times 1 = 2\sqrt{x}\)
• \(-5 \times 3\sqrt{x} = -15\sqrt{x}\)
• \(-5 \times 1 = -5\)

Each of these products represents one part of the multiplication process. By using the distributive property, we efficiently combine these into a single expression that is easier to manage.

Simplifying Expressions

Once we've utilized the distributive property to expand a multiplication problem, our next goal is to simplify each part of the expression. Simplifying means making the numbers and terms as concise as possible without changing their value.For example, one of the steps is simplifying \( (2\sqrt{x})(3\sqrt{x}) \). This involves multiplying the coefficients (2 and 3) to get 6. Then, since \((\sqrt{x})(\sqrt{x}) = x\), the entire expression simplifies to \( 6x \). Each piece of the expanded expression is evaluated similarly until all are as simple as possible:

• \(2\sqrt{x} \) remains as it is.
• \(-15\sqrt{x} \) is already simplified.
• \(-5 \) is also in its simplest form.

Carefully simplifying expressions ensures that the final result is neat and easy to read.

Combining Like Terms

After expanding and simplifying expressions, the next step is to combine like terms. Like terms are those that have the same variable raised to the same power. In our example, terms involving \(\sqrt{x}\) need to be combined.Starting from the simplified expression: \[6x + 2\sqrt{x} - 15\sqrt{x} - 5\]We group and simplify the terms involving \(\sqrt{x}\):

• \(2\sqrt{x} \) and \(-15\sqrt{x}\) are like terms. Combining them yields \((-13)\sqrt{x}\).

After combining these, the expression reads:\[6x - 13\sqrt{x} - 5\]By combining like terms, we simplify the problem further which helps us reach the clearest, most manageable form of the expression.