Problem 53 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (2 \sqrt{x}-5)(3 \sqrt{x}+1) $$

Step-by-Step Solution

Answer

The product simplifies to \(6x - 13\sqrt{x} - 5\).

Step 1: Expand the Expression

Use the distributive property to expand the expression \((2\sqrt{x} - 5)(3\sqrt{x} + 1)\). This involves multiplying each term in the first parenthesis by each term in the second parenthesis.Calculate:- \(2\sqrt{x} \cdot 3\sqrt{x} = 6x\)- \(2\sqrt{x} \cdot 1 = 2\sqrt{x}\)- \(-5 \cdot 3\sqrt{x} = -15\sqrt{x}\)- \(-5 \cdot 1 = -5\)

Step 2: Combine Like Terms

Combine the terms obtained from the expansion. Add \(2\sqrt{x}\) and \(-15\sqrt{x}\) (both are like terms):\(6x + 2\sqrt{x} - 15\sqrt{x} - 5\)= \(6x - 13\sqrt{x} - 5\).

Step 3: Simplified Expression

The expression \(6x - 13\sqrt{x} - 5\) is in its simplest form, as there are no further like terms to combine or simplify.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows us to simplify expressions and solve equations. It helps in expanding expressions where one term is multiplied with the entire sum or difference of other terms. The rule is expressed as: \(a(b + c) = ab + ac\). It means you take the term outside the bracket and multiply it by each term inside the bracket.
This principle is crucial when dealing with expressions involving terms like \((2\sqrt{x}-5)(3\sqrt{x}+1)\). Here, each term in the first parenthesis multiplies with every term in the second.

• So, when using the distributive property, ensure that each term in the first group is applied to each term in the second group.
• This will help you correctly expand complex expressions.

Understanding this property can make more advanced algebra topics easier.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. It is like cleaning up the equation to highlight the essential elements without any unnecessary clutter.
Once you have expanded an expression using the distributive property, the next step is simplifying it, which might involve:

• Combining like terms
• Rewriting radicals in a more straightforward form

When you simplify \(6x + 2\sqrt{x} - 15\sqrt{x} - 5\), the expression turns into \(6x - 13\sqrt{x} - 5\). This step helps make solving math problems more efficient and less error-prone.

Like Terms

Like terms in an algebraic expression are terms that have identical variables raised to the same power. They can be combined by adding or subtracting their coefficients. This is a vital step in simplifying expressions.
For instance, in the expression \(2\sqrt{x} - 15\sqrt{x}\), both terms are like terms because they contain the same variable \(\sqrt{x}\). You can combine these by adding or subtracting the numbers in front. Here,

• Subtracting gives us \(-13\sqrt{x}\)

This operation helps in reducing the complexity of expressions and is often necessary to arrive at the final simplest form.

Multiplication of Radicals

Multiplication of radicals involves specific rules to ensure accurate solutions. When multiplying square roots, you multiply the numbers inside and keep the square root sign.
In expressions like \(\sqrt{x} \times \sqrt{x}\), the result is \(x\), since the square root and the square are inverse operations. Understanding this can greatly simplify calculations involving radicals.

• When multiplying different radicals, multiply the values inside the radical signs and simplify the result if possible.

This knowledge is essential in dealing with expressions like \(2\sqrt{x} \cdot 3\sqrt{x} = 6x\). Mastering the multiplication of radicals can simplify solving algebraic expressions significantly, leading to clearer solutions.